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!NDUSTR1AL DRAWING AND 

GEOMETRY 



HENRY J. SPOONER. C.E. 




Qass \ ^S ^ 

Book ' ^^5 



INDUSTRIAL DRAWING AND GEOMETRY 



\ 



BT THE SAME AUTHOR. 



MACHINE DESIGN, CONSTRUCTION AND 
DRAWING : a Text-Book for the use of Young 
Engineers. With 126 Tables and over 1,600 Figures. 
8vo, los. 6 J. net. 

MACHINE DRAWING AND DESIGN FOR 
BEGINNERS : an Introductory Work for the use of 
Technical Students. Crown 4to, 3J. 61/. 

THE ELEMENTS OF GEOMETRICAL DRAW- 
ING: an Elementary Text-Book on Practical Plane 
Geometry, including an Introduction to Solid Geo- 
metry, etc. Crown 8vo, 3J. 6J. 

NOTES ON, AND DRAWINGS OF, A FOUR- 
CYLINDER PETROL ENGINE. Arranged for 
use in Technical and Engineering Schools. With 1 1 
Plates. Imperial 4to, zr. net. 

LONG.MANS, GREEN, AND CO., 

LONDON, NEW YORK, BOMBAY, AND CALCUTTA. 



INDUSTRIAL DRAWING AND 

GEOMETRY ' 

AN INTRODUCTION TO VARIOUS BRANCHES OF TECHNICAL DRAWING 



HENRY J. SPOONER, C.E. 

M.I.Mech.E., A.M.Inst.C.E., M.Inst.A.A., F.G.S., Hon.M.J.Inst.E., Etc., Etc. 

DIRECTOR AND PROFESSOR OF MECHANICAL AND CIVIL ENGINEERING IN THE POLYTECHNIC SCHOOL OF ENGINEERING, REGENT STREET, W. 

AUTHOR OP 

"MACHINE DESIGN, CONSTRUCTION AND DRAWING," "MACHINE DRAWING AND DESIGN FOR BEGINNERS'' 

'•THE ELEMENTS OF GEOMETRICAL DRAWING,"' "PRACTICAL PLANE AND SOLID GEOMETRY," "MOTORS AND MOTORING" 

"NOTES OK, AND DRAWINGS OF, A FOUR-CYLINDER PETROL ENGINE,'' ETC., ETC. 

WITH 620 FIGURES AND SiO EXERCISES 




LONGMANS, GREEN, AND 

39 PATERNOSTER ROW, LONDON 

NEW YORK, BOMBAY, AND CALCUTTA 

I9II 

All rights reserved 



CO. 






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i 



PREFACE 



Many years have passed since the Eev. Canon Moseley, the great Educationist, in reporting to the G-overnment on the importance 
of Geometrical Drawing as a Branch of National Education, wrote that "the use of scale and compasses in Drawing is a useful 
acquirement for workmen in almost every handicraft — to the blacksmith, the carpenter, and the mason, for instance, in making 
plans and sections of their work ; and to the gardener or agricultural labourer, in laying out a plot of ground to a scale, or planning 
a shed or a cottage, or the drainage of a field. It is, besides, a useful expedient of practical Education, to associate the conception 
of form with accurate linear dimensions, as is done in Geometrical Drawing. This kind of drawing might easily he taught to the 
first classes in Boys' Schools. The first step in it would probably be to make them copy drawings of the plans and sections of 
familiar objects ; then to practise them in making, to a scale, drawings of such plans and sections from the objects themselves ; 
lastly, the inventive faculty might (even in the case of such youthful scholars) be exercised by requiring them to design, and to 
draw to a scale, the plans and sections of simple and useful objects, such as are made by the carpenter, the mason, the blacksmith, 
or the plumber. 

" Eamiliaiized with the drawings of such plans and sections, they would hereafter be enabled to work from the latter with greater 
ease and correctness ; and that hahit of rtasoning and undei'standiiig about what they are xuorking iipon, which is the education of 
the vjorkman, would he encouraged and promoted. 

" Model Drawing, which your Lordships have so much and so beneficially encouraged, is, for the practical man, a subsequent 
step to the Geometrical Drawing of which I speak ; and however important and valuable in itself — if it be estimated by the 
extent to whifeh it is used or, indeed, useful in the mechanical arts — it must give place to it." 

It is quite believable that the above sagacious Report (the italics in which are mine) has had not a little to do with the very 
general inclusion of Geometrical and Mechanical Drawing in the curricula of our Elementary and Secondary Schools. The 
educational value of the subject is now well established ; for in addition to being a universal language and a means of expression, 
it is closely allied and associated with pure Geometry and calculation, and is a powerful means of cultivating the inventive faculty 
and habits of thought and reflection ; Ijut these points are now well understood, and need not be further pui-sued. 

Now there are many books published on practical Geometry, and many on various kinds of Technical Drawing, but I do not 
know of one that comprehensively embraces the two in a form suitable for beginners, so, in writing this little book I have attempted 
to produce an up-to-date and introductory work for beginners, employing modern expedients in giving effect to Moseley's recom- 
mendations ; and in doing so have kept in view how necessary it is for the pupil to make a good start by first giving attention 
to the selection and workmanlike manipulation of the simple instruments and materials used for such work ; and for the guidance 



vi PREFACE 

of those working without a teacher I have included a set of six half-tone prints, made from photographs showing the actual 
operations, which expedient I believe was first used in my advanced work, " Machine Design and Drawing." 

I have explained how fairly hard pencils, sharpened in the right way, should be used to practise drawing different kinds of 
lines of good quality in various directions on the paper ; how circles and arcs can be neatly drawn so as to make proper contact 
with one another and with straight lines, to ensure neatness and precision in execution. These simple operations should be per- 
formed again and again before more difficult work is taken in hand ; as slovenly habits of drawing once acquired are extremely 
difficult to correct. A glance over the page of contents will show that a comprehensive, and in some chapters an unusual but useful 
selection of matter has been made for treatment in what is well-nigh an inexhaustible subject. Many of the drawings relate to 
the work of the architect, bricklayer, carpenter, engineer, industrial artist, mason, and metal-plate worker ; but apart from these, 
attention is called in suitable places to the application of geometry to a wide range of industrial work, including engraving, 
gardening, land surveying, lithography, optical work, printing, stereotyping, etc., etc. 

Most teachers, after giving a lecture, like to test the knowledge of their pupils by asking them pertinent questions ; bearing 
this in mind, I have given at the end of most chapters a few suggestive or typical oral questions, followed by sketching and 
drawing exercises. 

The order in which I have arranged the chapters seemed to me to give, on the whole, the best sequence, but of course teachers 
can vary this very considerably in accordance with their own views. 

To make the book as attractive as possible to those who may use it without the help of a teacher, I have, when convenient, and 
whenever the matter treated seemed to lend itself to it, adopted a conversational style. 

I am hoping that the little work will meet the requirements of many who are teaching the subject in Elementary, Secondary, 
and Trade Schools, and that it will conveniently lead up to works on Machine Construction and Drawing (such as my " Machine 
Drawing and Design for Beginners "), on Building Construction, and other branches of Technical Drawing. 

I may add that the London University Matriculation Syllabus in Geometrical and Mechanical Drawing is nearly covered by 
the contents of the book. 

HENEY J. SPOONEE. 
The Polytechnic School of ENQiNEEEiNa, 
Eegent Street, London, W.- 
August, 1911. 



CONTENTS 



CHAP. PAGE 

I. Drawing Instruments, Materials, etc., and their Use .... 1 

II. How to draw Straight Lines and Simple Figures 7 

III. Measurement and Construction of Angles, etc 14 

IV. Construction o£ Triangles 18 

V. Scales, their Construction and Vse 22 

VI. Proportion, etc ' 27 

VII. Circles, Arcs, and Lines 32 

VIII. Use of Squared or Sectional Paper 37 

IX. Areas and their Measurements 42 

X. Eeducing and Enlarging Figures, etc 57 

XI. Symmetry and Symmetrical Figures 62 

XII. The EUipse 65 

Xni. The Parabola 73 

XIV. The Hyperbola 78 

XV. Spirals and Miscellaneous Curves 81 

XVI. The Application of Geometry to Ornamental and Decorative 

Design 87 



CHAP. PAGE 

XVII. Plan and Elevation — How to make a Working Drawing of a 

SoUd Body 94 

XVIII. Projections and Sections of some Typical Solids 102 

XIX. Further Studies in Projection Ill 

XX. Isometrical Projections and Drawings 116 

XXI. Oblique or Pictorial Drawing 120 

XXII. Simple Fastenings used in Metal Work, and how to Draw 

them 124 

XXIII. Miscellaneous Drawing Exercises in Woodwork, Brickwork, and 

Masonry 132 

XXIV. Various Machine-Drawing Exercises 137 

XXV. Intersections and Development of Simple Bodies 145 

XXVI. Printing, Shading, Tracing, etc 152 

XXVII. Miscellaneous Drawing Exercises 157 

Definitions, etc • 160 

Tables of British and Metric Equivalents 169 

Miscellaneous Constants 169 



DETAILS OF CONTENTS 



. CHAPTER I 

DRAWING INSTRUMENTS, MATERIALS, ETC., AND 
THEIR USE 

PACK 

1. Compasses, Dividers, Pens, etc. 2. Drawing Board. 3. Working 
Position of Drawing Board. 4. T-Square. 5. Set-Squares, or 
Triangles. 6. Using Set-Squares for Parallel Lines. 7. Drawing 
Paper. 8. Whatman's Papers. 9. Pencils — Different Kinds and 
Qualities. 10. How to Sharpen the Pencil. 11. Compass Pencils. 
12. The Conical-pointed Pencil. 13. Drawing Pins, 14. The Rule, 
and taking Measurements from it. 15. Drawing or Ruling Pens. 
16. Indian Ink. 17. Colours, etc. 18. Saucers for mixing Colours. 
19. Brushes 1-6 

CHAPTBE II 

HOW TO DRAW STRAIGHT LINES AND 
SIMPLE FIGURES 

20. Introduction. 21. Example — Straight Lines drawn with the Assist- 
ance of the T-Square. 21a. Defects in Lines. 22. Straight Lines 
drawn with the Assistance of a Set-Square. 23. Dotted Lines. 24. 
Rectangles. 25. To draw a Rectangle of given Length and Breadth. 
26. Exercises upon the Use of Centre Lines : Figure Sy mm etrical 
about a Single Centre Line. 27. Figure Symmetrical about Two 
Centre Lines. 28. Another Example. Drawing Exercises .... 7-13 



CHAPTEE IV 

CONSTRUCTION OF TRIANGLES 

r.vuK 
37. Definitions, etc. Congruent Triangles. 38. To construct a Triangle 
with Sides any given Lengths. 39. To construct an Equilateral 
Triangle. 40. To construct a Triangle, the Angles being in a given 
Proportion. 41. To construct a Triangle, a Side, an Angle at the 
■ Side and the Perimeter being given. 41a. To construct a Triangle, 
the Base, the Vertical Angle and Altitude being given. 42. To con- 
struct a Eight-Angled Triangle, having given the Hypotenuse and 
One of the Acute Angles. 43. To construct a Triangle of given 
Perimeter, whose Sides are in a given Proportion. 44. To draw the 
Inscribed and Circumscribing Circles of any Triangle. Exercises 18-21 

CHAPTER V 

SCALES, THEIR CONSTRUCTION AND USE 

45. Introduction. 46. Drawing to Scale. 47. Engineer's Scales. 48. 
Construction of Scales. 49. To divide a Line into any Number of 
Equal Parts. 50. To draw a Scale of j'j, to read Feet and Inches. 
51. To draw a Scale of l|" to 1 yard, to show Feet. 52. To draw a 
Centimetre Scale. 53. Diagonal Scales. 54. To construct a Diagonal 
Scale to show Eighths and Sixty-fourths of an Inch. 55. To draw a 
British and Metric Conversion Scale. Use of the Scale. Exercises 22-26 



CHAPTER III 

MEASUREMENT AND CONSTRUCTION OP ANGLES, ETC. 

29. Introduction. 30. To draw, with the Assistance of the Protractor, a 
Line making a given Angle. 31. The Use of Set-Squares in setting 
out Angles. 32. Use of Set-Squares in Trisecting and Bisecting 
Angles. 33. Copying Angles. 34. The Fitter's Square. 35. The 
Pliimh Rule. 86. The Spirit Level. Exercises 14-17 



CHAPTER VI 
PROPORTION, ETC. 

56. Introduction. 57. To divide any Line into Three Parts in a given 
Proportion. 58. To divide a Line into Two Parts in a given Pro- 
portion. 59. To divide a Line Proportionally to any given Divided 
Line. 60. To divide the Space between Two Parallel Lines into 
Equal Parts by Lines Parallel to them. 61. To find a Third Pro- 



DETAILS OF CONTENTS 



portional to Two given Lines. 62. To find a Fourth Proportional to 
Three given Lines. 63. To find the Height of a Tower from its 
Shadow. 64. To find the Mean Proportional to Two given Lines. 
Exercises, Oral and Drawing 27-31 



CHAPTEE VII 

CIRCLES, ARCS, AND LINES 

65. Introduction. 66. Definitions. 67. To describe a Circular Arc 
through Three given Points. 68. To find the Centre and Radius of 
a given Arc or Circle. 69. To draw a Tangent to a Circle through a 
fixed Point in its Circumference, 70. To draw a Tangent to a Circle 
through a fixed Point without it. 70a. To inscribe in a given Angle 
a Circle of given Radius. 70b. To describe a Circle of given Radius 
to touch a given Line and a given Circle. 71. To describe an Arc of 
a Circle of given Radius to touch a given Arc and a given Straight 
Line. 72. To describe an Arc of a Circle to touch a given Line iu 
a fixed Point and also a given Arc. 73. To draw a Circle to touch 
Three given Straight Lines. 74. To describe Two Arcs to meet each 
other in the Line of their Centres and to touch Two given Lines at 
fixed Points in them, the Radius of the lesser Arc being also given. 
75. Interesting cases of Arcs flowing into one another. 76. Hints 
on working the Exercises. Exercises 32-36 



CHAPTEE VIII 

USE OE SQUARED OR SECTIONAL PAPER 

77. Introduction. 78. Different kinds of Squared Paper. 79. Use of 
Squared Paper. 80. Position of a Point in a Plane. 81. Position 
of a Straight Line in a Plane. 82. To draw a Graph. 83. Straight 
Lines that do not pass through the Origin, and their Equations. 
84. Position of a Rectilinear Figure in a Plane. 85. Irregular 
Polygon fixed in a Plane by its Co-ordinates. 86. Plotting Ex- 
perimental Results on Squared Paper. Oral Questions. Drawing 
Exercises 37-41 



CHAPTEE IX 

AREAS AND THEIR MEASUREMENTS 

87. Introduction. 88. To construct a Rectangle equal in Area to any 
given Parallelogram. 89. To draw an Isosceles Triangle whose Area 



shall be half that of a given Rectangle. 90. To construct a Rect- 
angle equal in Area to a given Triangle. 91. To construct a Rect- 
angle equal in Area to a given Rectangle, One Side of the former 
being given. 92. The Pythagorean Theorem. 93. To construct a 
Square equal in Area to the Sum of Two given Squares. 94. Area 
of a Circle. 95. To construct a Rectangle equal in Area to a given 
Circle. 96. To describe a Circle equal in Area to the Sum of Two 
given Circles. 97. To draw a Square whose Area shall equal, say. 
Five Square Inches. 98. To malie a Square equal in Area to a given 
Rectangle. 99. To reduce any Irregular Figure to a Triangle of 
equal Area. 100. Measurement of Land. 100a. An Acre. 101. 
The Land Surveyor's Operations. 102. The Field-book. 103. To 
measure a Field of Three Sides. 104. To measure a Field with 
several Sides. 105. Offsets, and how to measure them. 106. To 
find the Area of a Rectangvilar Enclosure in Square Feet. 107. To 
draw a Square or any Regular Figure equal in Area to a Closed 
Figure bounded by an Irregular Curved Line. 108. To find the Area 
of a Figure bounded by Three Rectangular Lines and an Irregular 
Curved Line. 109. Anothermethodof measuring Areas. 110. Given 
a Closed Figure bounded by an Irregular Curved Line, representing 
an Indicator Diagram from a Steam Engine, to determine its Ap- 
proximate Area in Square Inches. 111. Areas of Figures on Squared 
Paper, (a) When a Side or Diagonal of the Figure can be made to 
coincide with a Line of the Squared Paper. (6) When an important 
Line of the Figure does not coincide with a Line of the Squared 
Paper, (c) When the Figure is Curvilinear. Typical Oral Questions. 
Drawing Exercises 42-56 



CHAPTEE X 

REDUCING AND ENLARGING FIGURES, ETC. 

112. Introduction. 113. Proportional Compasses. 114. To reduce or 
enlarge the Lines of a Figure, using Proportional Compasses. 115. 
To reduce a given Irregular Figure to a similar one whose Sides 
shall be, say, half of those of the given one. First Method ; Second 
Method. 116. Variations of the Previous Methods. 117. To reduce 
a given Figure bounded by a Curved Line to a similar one of Fixed 
Size. 118. Reducing and Enlarging Figures by the Use of Squares. 
119. To construct a Figure similar to a given Figure but with Twice 
its Area. 120. To construct a Figure similar to a given Figure, and 
having an Area equal to that of another given Figure. 121. The 
Mass-centre or Centre of Area or Gravity. 122. To find the Mass- 
centre of any Quadrilateral Figure. Alternative Method. Drawing 
Exercises 57-61 



DETAILS OF CONTENTS 



XI 



CHAPTER XI 
SYMMETRY AND SYMMETRICAL FIGURES 

PAGE 

123. Introduction. 124. Centre of Symmetry. Axes of Symmetry. 

125. Use of Squared Paper in Dravring Symmetrical Figures. 

126. Practical Applications of Symmetrical Figures — Engraving, 
Lithography, and Printing. 127. Steoreotyping. Oral Exercises. 
Drawing Exercises 62-64 



CHAPTEE XII 

THE ELLIPSE 

128. Introduction. 129. Some Definitions and Properties of the Curve : 
Diameter, Vertices, Transverse Axis, Conjugate Axis, Focal Dis- 
tances, Tangent, Normal, Conjugate Diameter, Area. 130. To draw 
an Ellipse (First Method) as the Locus of a Point. 181. To draw an 
Ellipse, having given the Major and Minor Axes (Second Method) : 
by Concentric Circles. 132. (Third Method) : Mechanically, by means 
of a piece of String and Two Pins. 138. (Fourth Method) : by Paper 
Trammel. 184. (Fifth Method) : by Radial Lines from the Ends of 
the JNIinor Axis. 185. To determine the Major and Minor Axes of an 
Ellipse. 136. To draw a Tangent to a given Ellipse at a Fixed Point 
in the Curve. 137. To draw a Normal or Perpendicular to a given 
Ellipse at a Fixed Point in the Curve. 138. To draw an Ellipse 
when Two Conjugate Diameters, other than the Major and Minor 
Axes, are given intersecting each other at their Centres. (First 
Method) : by Radial Lines from the Ends of One of the Axes. 139. 
(Second Method) : by using the Auxiliary Circle. 140. Approximate 
Method of constructing Ellipses by Circular Arcs, having given the 
Number of Centres. First Method : (by Three Centres), when the 
Major and Minor Axes are given. 141. Second Method (another 
Method by Three Centres), when the Major and Minor Axes are given. 
142. Third Method (by Three Centres) : when the Minor Axis is not 
less than Two-thirds the Major Axis. Typical Oral Exercises. 
Drawing Exercises 65-72 



CHAPTER XIII 

THE PARABOLA 

143. A Parabola. 144. Definitions and Properties, etc. : Diameter, 
Axis, Vertex, Principal Vertex, Ordinate, Abscissa, Chord, Tangent, 
Sub-tangent, Normal, Sub-normal. 145. Diameter and Focal Line, 
Parabolic Reflector. 146. To draw a Parabola when its Axis and 



Base are given. First Method. 147. Second Method : by Tangents 
Bending Moment Diagrams. 148. To describe a Parabola when its 
Base, Height, and InoUned Axis (which passes through the Centre of 
the Base) are given. 149. To describe a Parabola when the Directrix 
and Focus are given. Typical Oral Exercises. Drawing Exercises 78-77 



CHAPTER XIV 

THE HYPERBOLA 

150. Introduction. Asymptotes, The Rectangular Hyperbola. 151. Given 
an Ordinate and Abscissa of a Point in a Rectangular Hyperbolic 
Curve, and the Axes (or Asymptotes) to draw the Curve. Typical 
Oral Exercises. Drawing Exercises 78-80 



CHAPTER XV 

SPIRALS AND MISCELLANEOUS CURVES 

152. Introduction. 153. The Involute of the Circle. 164. To describe 
the Involute of a Circle. 154a. Cams. 155. To set out a Cam giving 
a slow upward, an interval of rest, and a quick downward motion to 
a Slider. 156. To set out a Simple Cam. 157. Involute Cams. 
158. The Spiral of Archimedes. 159. To describe One Convolution 
of the Spiral of Archimedes, the Pole, the Initial Line, and the 
length of the longest Radius Vector being given. 160. The Curve 
of Sines, or Harmonic Curve. 161. To draw the Curve of Sines, the 
Amplitude and Length of a Vibration being given. 162. The Helix. 
163. Screw Threads are Helices. 164. Plotting Paper for Polar 
Co-ordinates. Typical Oral Exercises. Drawing Exercises . . 81-8G 

CHAPTER XVI 

THE APPLICATION OF GEOMETRY TO ORNAMENTAL AND 
DECORATIVE DESIGN 
165. Introductory Remarks. 166. Marquetry, or Buhl Work. 167. 
Mosaic. 168. Some Points relating to the Application of Simple 
Figures in Mosaic— Paving, Glazing, and generally to aU Inlaid 
Work and Geometrical Patterns. 169. Simple Patterns formed by 
Equilateral Triangles. 170. To draw a Diamond Chequered Pattern. 
Cast-iron Grating. 171. Simple Star Forms. 172. Patterns suit- 
able for Tiles, Linoleum, etc. 178. Greek Frets. 174. Trellis Pat- 
terns. 175. Hexagons in a Rectangle. 176. Use of Guiding Lines 
in Detail Work. Typical Oral Exercises. Drawing Exercises . 87-93 



DETAILS OF CONTENTS 



CHAPTEE XVII 

PLAN AND ELEVATION— HOW TO MAKE A WOEKING 
DRAWING OP A SOLID BODY 

TAGK 

177. Introduction. 178. Plan and Elevation. 179. To draw tlie Plan 
and Elevation of a Rectangular Block, with a Pace horizontal, and 
at a given Height, and One of its Sides parallel to the Vertical Plane. 
180. End (or Side) Elevations, and Sections. 181. Drawings of a 
Cast-iron Bench Block. Typical Oral Exercises. Drawing Exercises 94-101 

CHAPTEE XVIII 

PROJECTIONS AND SECTIONS OP SOME TYPICAL SOLIDS 

182. Introduction. 183. To draw the Plan and Elevation of a Rect- 
angular Block, with its Base inclined 30° to the Horizontal, and one 
of its Sides parallel to the Vertical Plane. 184. To draw the Plan 
and Elevation of a Hexagonal Pyramid when its Base is on the 
H.P., with an Edge of the Base inclined to the V.P. 18S. To draw 
the Projections of a Tetrahedron when one of its Slant Edges is 
parallel to the V.P., and a Pace is on the H.P. 186. A Cylinder 
with its Base on the H.P. is cut by a Plane passing through the Top 
Left-hand Corner of its Elevation, and making an Angle of 60° with 
its Axis. To draw Plan and true Shape of Section. 187. Conic 
Sections. Cone cut giving the following Sections ; Triangle, Circle, 
Hyperbola, Ellipse, and Parabola. 188. Given a Cone with its Base 
on the H.P., to draw the true Shape of a Section made by a Cutting 
Plane. First case ; by a Plane parallel to its Axis. 189. Second 
case : by a Plane inclined to the Axis at a given Angle and passing 
through a Point in the Base. 190. Third case : by a Plane parallel 
to its Side. 191. Preliminary Projections. 192. To draw Plan and 
Elevation of a Cylindrical Roller, when a Pace is inclined to the 
V.P. 193. To draw the Plan and Elevation of a Cone when lying on 
its Side. 194. A Horizontal Line parallel to the V.P. is given by 
its Plan and Elevation, to determine the Distance from the XY. 
Drawing Exercises . 102-110 

CHAPTEE XIX 

FURTHER STUDIES IN PROJECTION 

195. First-angle versus Third-angle Projection, or English versus Ameri- 
can Practice. 196. End Elevations or Side Views ; best arrangement 
of views. 197. To draw a Section of a Wrought-iron or Steel Beam. 
198. British Standard Beam Sections. 199. To draw Three Views of 
a Hook Bolt. 200. To draw a Stuffing Box Gland, 201. To draw 



the Plan and Elevation and Section of a Steel Crank. 202. Use of 
Pictorial Sketches and Drawings 111-115 

CHAPTEE XX 

ISOMETRICAL PROJECTIONS AND DRAWINGS 

203. Introduction. 204. The Difference between Isometric Projection 
and Isometric Drawing. 205. Some Typical Isometric Drawings. 
206. Isometric Drawing of a Circle. 207. Setting off Angles to the 
Sides of the Isometric Cube. Typical Oral Exercises. Drawing 
Exercises 116-119 

CHAPIEE XXI 

OBLIQUE OR PICTORIAL DRAWING 

208. Introduction. 209. Some Simple Bodies drawn in Oblique Projec- 
tion, Drawing Exercises 120-123 

CHAPTEE XXII 

SIMPLE FASTENINGS USED IN METAL WORK, AND 
HOW TO DRAW THEM 

210. Introduction. Riveted Joints. 211. Proportions of Rivet Heads. 
212. Forms of Joints. 213. Double Riveted Lap Joints. 214. 
Proportions of Joints. 215. Diameter of Rivets. 216. Pitch of the 
Rivets. 217. Butt Joints with Double Straps. 218. Hints on 
Making the Drawings. 219. Screws, Bolts, etc. 220. Forms of 
Screw Threads. 221. Proportions of Screw Threads. 222. To draw 
a 1" Whitworth Bolt and Nut. Proportions of Standard Whitworth 
Bolts and Nuts. 223. Standard Bolts and Screws. 224. Locking 
Nuts and Arrangements. 225. The Capstan Nut or Castle Nut. 
Typical Oral Exercises. Sketching Exercises. Drawing Exercises 124-131 

CHAPTEE XXIII 

MISCELLANEOUS DRAWING EXERCISES IN WOODWORK, 
BRICKWORK, AND MASONRY 

226. Introduction. 227. Wooden Stand for a Machine. 228. Kitchen 
Table. 229. Dining-room Sideboard. 230. Brickwork and Masonry. 
231. A 9" Brick WaU. English Bond and Flemish Bond. 232. A 
14" Brick Wall. 233. A small Church. 234. Angle Quions with 
Bevelled Edges. 235. Stone Semicircular Arch. 236. Segmental 
Arch of a Stone Bridge. 237. Square-headed Window with Flat- 
gauged Arch 132-136 



DETAILS OF CONTExNTS 



xui 



• CHAPTBE XXIV 

VAEIOrS MACHINE-DRAWING EXERCISES 

238. Cast-iron Bracket. 239. Cast-iron Foundation Washer. 240. 
Engine Fitter's Square. 241. Gun-metal Flanged Bush. 242. 
Link End with Spherical-seated Bush. 243. Joint Pins. 244. Belt 
Pulley. 245. Locomotive Crank. 246. Cast-iron Bearing Block. 
247. Gun-metal TaU-guide for Valve Rod. 248. Couplings. 249. To 
draw a Butt-MuS Coupling. Proportions of Couplings, Materials, 
etc. 249a. Drum or Barrel of Hoisting Machine. 250. How to 
measure the Diameter of a Large Cylindrical Body. 251. Petrol 
Engine Piston. 252. Petrol Engine Cormecting Rod. 253. Cast- 
iron Bracket with Pin 137- 



144 



CHAPTEE XXV 

INTERSECTIONS AND DEVELOPMENTS OF SIMPLE BODIES 

254. Intersections. 255. Intersection of Two Equal Cylinders. 256. 
Intersection of Cylinder and Cone. 257. Intersection of Two Cones 
and a Cylinder. 258. Intersection of Two Unequal Cylinders. 259. 
Intersection of Circular Fillet and Plane Surface. 260. Intersection 
of Circular FiUet with CyUndrical Surface. Developments. 261. 
Introduction. 262. Development of the Five Regular Solids. 263. 
Development of a Square Elbow. 264. By Direct Method. 265. De- 
velopment of a Square Tee Piece. 266. Development of a Cone. 
267. Developments of Pyramids. Exercises 145-151 

CHAPTEE XXVI 
PRINTING, SHADING, TRACING, ETC. 
268. Printing, Lettering, etc. 269. Workshop Drawings. 270. Shade 



Lines. 271. Shading by Lines, Surfaces in the Light, Surfaces in 
the Shade. 272. Copying Workshop Drawings. Blue Prints. 273. 
Tracing. 274. Tracing Exercises. 275. Sectional Shading or Lining 
for Various Materials 152-1 56 



CHAPTEE XXVII 

MISCELLANEOUS DRAWING EXERCISES 

Drawing of a Hut. Dravrings of a Cast-iron Weight. A Wooden Box. 
Part of a Gun-metal Thrust Bearing. A Dovetailed Joint. A Gun- 
metal Neck Bush for a Stuffing Box. Valve Cams. A Gun-metal 
Gland. Bracket Bearing for Cam Shaft. Petrol Engine Valve 
Cover. Cast-iron Flange Joint. Petrol Engineer Cylinder Cover 
Plate 157-159 



DEFINITIONS, ETC. 

Definitions relating to Plane Figures. Definitions and Summary of 
some Useful Particulars relating to Areas. Definitions, etc., relat- 
ing to Solids and their Projection. Surfaces and Volumes of the Five 
Regular Solids 160-168 



TABLES OF BRITISH AND METRICAL EQUIVALENTS 

Length. Land Measure. Surface and Area. Volume 169 

Miscellaneous Constants 169 



"T^Mri^^. 



''"^"'^Q^' 



' ""-H. 



INDUSTRIAL DRAWING AND GEOMETRY 



'/ CHAPTER I 

DRAWING INSTRUMENTS, MATERIALS, ETC., AND THEIR USE 

Introduction. — A few of the instruments and things described in this chapter you may not elect to use for some time to come ; 
indeed, if you are doing only pencil- work, you will not require them ; but even so, you will probably like to know something about 
these things, and it will not be a waste of time to quietly look through the chapter at your leisure. 

1. Compasses, Dividers, Pens, etc. — A few shillings will now buy a small set of well-made drawing 
instruments of the English type, with which much useful work can be done. Usually these sets contain a 
pair of compasses with pen and pencil points, a pair of dividers, a bow-pencil, a bow-pen, and a drawing 
or ruling pen. Of course these instruments are not to be compared with the heavier and better made ones 
turned out by the best English makers, which you will be anxious to provide yourself with later. 

2. Drawing Board. — This instrument is used for holding and supporting a sheet of paper flat, whilst a 
drawing is being made upon it. Care should be exercised in its selection, or trouble may be occasioned by 
its becoming twisted and out of truth, after very little use. There are many kinds of drawing boards, but 
the "Battened" form, shown in Fig. 1, is the best. The size most suitable for exercise work is about 
24" X 17'7 which takes the half of an " imperial " sheet of paper, or a " medium " sheet. 

3. Working Position of Drawing Board. — To enable you to get a good view of your work without leaning 
over too much, the drawing board when in use should be tilted to an angle of about 15°,^ either by using it 
on a sloping desk, or with the aid of wooden blocks. 

4. T-Square. — This instrument is used for drawing long lines perpendicular to an edge of the drawing 
board ; and Eig. 2 shows the " English shape," which is best for general purposes. It is made of well- 
seasoned pearwood, maple, or mahogany. Those of pearwood are the cheapest and answer very welL'for 

" The mark or sufHx (") signifies inch or inches : thus -['j" reads — one-sixteenth of an inch. A singlei dash (') signifies foot or 
feet : thus 16' reads — 16 feet. The sign x coming between the two dimensions 24 and 17 signifies multiplication or the word hy. 
Thus the above would read, 24 inches by 17 inches, one dimension being multiplied by the other giving the area of the board. 

- The sufSx (') signifies degrees : thus 15° reads — 15 degrees. Degrees are divided for measuring purposes into 60 equal parts, 
called minutes, represented by the suffix (') ; and minutes into 60 parts CEilled seconds, represented by ("). Thus 15° — 30' — 40" would 
read fifteen degrees, thirty minutes, forty seconds. 

1 




Fig. 1. — Battened draw- 
ing board. 
B 



0^^ 



2 INDUSTRIAL DRAWING AND GEOMETRY 

rough use in a school, but the raahogauy ones, with the workhig edges of ebony, are generally used for office work, and should 
always be used by those who can afford them. An enlarged section of the ruling edge is shown at A on Fig. 2. 







Pig. 2. — English shape T-square. 



Pig. 3. — Testing T-square. 



Pig. 4. — The two set-squares. 



5. Set-Squares, or Triangles.^ — These are right-angled triangles. They are made of various materials — such as pearwood, 
mahogany, and other woods, vulcanite, and transparent celluloid ; and are used for drawing short lines perpendicular to a straight 

edge, T-square, or another set-square. They are also used for 
drawing angles of 30°, 45°, and 60°. 

Two set-squares are generally used, the angles and most 
useful sizes for which are shown on Fig. 4; 

Set-squares of pearwood are cheap and useful for school 
use, but they are easily soiled, and often warp and become 
untrue. They are not to be compared with those made of trans- 
parent celluloid, which on the whole should he preferred. 

6. Using Set-Squares for Parallel Lines. — Parallel lines that 
cannot be drawn by using the T- and set-squares in the ordinary 
way may be drawn by sliding the set-squares on one another, as 
shown in Fig. 5, where the set-square A is held firmly on the 
paper, and the other B is slid along the edge ah. Three posi- 
tions, B, B', B", of the set-square B are shown ; the line ccl 
being parallel to the corresponding lines drawn through B and B', and perpendicular to the line ab. Thus by manipulating the 
squares in this way, lines parallel or perpendicular to one another can be drawn on any part of the paper. 

' The name given to these instruments in America. 



Fig. 5. 




-Using set-squares for 
parallel lines. 



Testing set-squares. 



DRAWING INSTRUMEiNTS, MATERIALS, ETC., AND THEIR USE 3 

7. Drawing Paper. — Two kinds of paper are ia general use for drawing purposes : viz. " Cartridge " imper and " Draiving " 
2iapcr. "Cartridge" (or "Machine-made") drawing paper is used for ordinary school drawing purposes. It is much cheaper than 
Whatman's " drawing paper " (which is used by engineers and architects), and can be obtained either in sheets or in rolls up to 
62" wide and 60 yds. long, rendering it extremely useful for diagrams, etc. Cartridge paper has two surfaces, a rough and a smooth 
one ; the smooth surface is the proper side to draw upon, and is usually the front side when the water-mark ^ can be read correctly on 
holding the sheet between the eyes and the light. 

Cartridge paper does not usually take tints of colour evenly, but with good paper, and care, a very fair effect can be obtained 
in light tints. But this paper is most suitable for line drawings. 

8. Whatman's Papers. — For drawings that are to be finished in ink, without colour, the " Hand-made " drawing paper known 
as Whatman's " Hot-pressed," H.P., " Smooth " or " Rolled " surface, is most suitable. This paper should also be used for drawings 
when very fine lines are a necessity, and but little colour is required. For drawings which are to be coloured or shaded, or are to 
stand frequent erasing of lines, Whatman's N.H.P. Paper {not hot-pressed) or rough surface is to be preferred ; its surface will take 
a fairly fine line, and tints can be laid very evenly upon it. 

9. Pencils — Different Kinds and ftualities. — You should, if possible, only use blacklead pencils of a good quality, such as Stanley's, 
Faber's, or Hardtmuth's ijrepared lead, or Cohen's Cximherland lead ; inferior makes are very unsatisfactory for drawing purposes. The 
following are the requirements of a good pencil for mechanical drawing : It should be moderately 

hard, of even colour throughout, and durable enough to retain a working point for a long time. B c d 

It should not be liable to roll off the board and injure its point, and the lines drawn by it r^\ /~m\ f 

should be easily rubbed out. The ordinary round cedar-covered blacklead pencil, shown at A, \_^ \ / <^^ • 

Fig. 7, of good quality, is a serviceable pencil, but it easily rolls off the board. To retard the -^^^^ 7.— Sections of blacklead pencils. 
rolling action, some pencils are made hexagonal (Fig. 7, B), whilst Messrs. Stanley & Co. sell 

a pencil of specially prepared lead, the wooden cover of which is made elliptical, as shown at C. Degrees of Hardness, etc.-- 
Pencils are made in various degrees of hardness, varying from BBBB (the softest) to HHHHHH (the hardest), and ISTos. 1 to 6 in 
the solid lead, and in some makes, such as Stanley's pencils. 

Usually No. 1 = BB. No. 2 = HB. No. 3 = H. No. 4 = HH, or 2H. No. 5 = HHH, or 3H. No. 6 = HHHH, or 4H. 
Nos. 3 and 4 will be found most useful for ordinary school work. The hardest pencils are only useful when of the very 
highest quality ; they are expensive, and are used for very fine work to a small scale. ^^__ c r 1 

10. How to Sharpen the Pencil. — For ordinary line drawing, the pencil should be ^ ^ ^ \ L^-J 

sharpened to a flat or chisel point, as shown in Fig. 8 ; this gives a strong point, which 

retains its sharpness longer than a round one, and it can be worked closer up to the 

squares, and is more easily sharpened ; with the added advantage that the lines are more 

equal in quality. Needless to say, it is used with its flat side laid against the edge of 

the T- or set-square. To make a flat or chisel point to a wood-covered pencil, the wood t? s r>,- 1 ■ t d u 

is first cut away ; and the best way to do this is to hold the pencil, as shown in Fig. 9, ^*'" '" ^^^ '^°™ ^ P ■ 

between the thumb and fir.=t finger of the left hand, and to rest it upon the second finger, which should be turned upwards, while 

' The best qualities only are water-marked. 




INDUSTRIAL DRAWING AND GEOMETRY 



the penknife (which should be sharp) is held in the four fingers of the right hand, which should be turned downwards, the thumb 
of this hand being placed under the pencil to steady it, as shown. A little practice will enable you to cut a good point with 





Pig. 9. — Knifing pencil point. 



Fig. 10. — Filing pencil point. 



precision and facility, as you have perfect control over the knife, which, should it slip, moves away from your hand. The lead part 
is best sharpened by rubbing it upon a smooth file,^ as shown in Fig. 10, after which a stroke or two upon a piece of smooth paper 
gives it a good finish. 

11. Compass Pencils. — The points of compass pencils should be made narrower than for straight-line purposes, and must be 
carefully adjusted so as not to draw a thick line ; indeed, the beginner is more likely to do better work with a conical-pointed 
lead in his compasses. It is not enough to start with a good point, its sharpness must be maintaiaed, and this requires constant 
attention. 

12. The Conical-pointed Pencil. — For the making of freehand sketches, dimensioning, or descriptive writing upon a pencil 
drawing, it is desirable to use a softer pencil than that used for line drawing (such as a ISTo. 2, or 3, or HB or H), and to sharpen 

it to a long conical point, as shown in Fig. 11. The point should on no 

i^f= m , account be moistened when used, as marks made by it in that condition are 

w very difficult to erase. 

13. Drawing Pins. — To secure the paper to the drawing board drawing 
pins are used. For common school use small stamped pins answer very 



Fig. 11. — Conical-pointed pencil. Fig. 12. — Drawing pin. 

well ; but a better, although more expensive, form is shown in Fig. 12. 



'■ A 4" smooth file, or a i" triangular or three-square saw file, should be preferred. If a file is not available, a piece of fine emery paper or cloth, 
" FF," or glass paper, " 0," fastened to a strip:of hard wood about 6" long, 1" wide, and a J" thick, is a good substitute, or small blocks, containing about 16 
of glass paper, specially made for pencil sharpening, can be obtained. 



F" or 
16 sm-faces 



DRAWING INSTRUMENTS, MATERIALS, ETC., AND THEIR USE 




Fig. 13. — Showing application of dividers to rule. 



14. Tie Rule/ and taking Measurements from it. — A 12-inch steel rule, divided to 64ths and lOOtlis of an inch, should be 
preferred. Dimensions can very conveniently be taken off it by the dividers, as in Fig. 13, and pricked off on the drawing. In 

doing this care should be taken not to place the points of the dividers upon _ 

the rule in a normal or upright direction, or they will be injured. Fig. 13 
shows how, by inclining the dividers to the surface of the rule, the sides of the 
points may be made to rest in the cuts or divisions without injuring the points 
or the divisions of the rule, if the latter be made of a soft material. The 
figure also shows how the dividers and rule should be held if the right hand 
is to have complete command over the former in adjusting the points to take 
off any required dimension. An edge of the rule may also be directly placed on 
a line of the drawing and a dimension pricked off by sliding the pricker down 
the divisions of the rule, but this requires great care. The accuracy of the steel 
rule and its durability make it superior to any other at the command of the 
draughtsman. As you may be frequently called upon to set out work with 
metric measurements, the back of the rule should be divided into centimetres 
and millimetres. You will also find a pair of calipers and a 60" measuring tape 
useful, as you will see later. 

15. Drawing or Ruling Pens are used in inking in drawings. The best 
type of these pens is jointed, so that when the screw is taken out one of the nibs can be moved away from the other about its 
hinge or joint for cleaning purposes. 

16. Indian Ink. — It is well known that ordinary writing ink is unsuitable for use on drawings, as, although it is more or 
less indelible,^ it has not the blackness and body that ai'e considered necessary, to say nothing of the corrosive action of such inks 
on steel, which alone would preclude its use in the ordinary drawing ijcn. In addition to these objections ordinary writing ink 
runs too freely from the pen and blurs when touched by a brush in colouring. The only ink that satisfies all the draughtsman's 
requirements is known as Indian ink ;^ this ink, which may be produced by grinding or rubbing down an ink stick, when properly 
used, produces a clean, dense, jet-black line, and, being free from acid, it does not corrode the instruments ; it can also be obtained 
in a liquid form. 

17. Colours, etc., for tinting drawings, may be obtained in cakes, or small pans, and very few suffice to begin with. If cake 
colours are used, they are ground up with water in a saucer until of the required depth of tint. Moist water colours in pans are 
to be preferred for school use. They are to be had in tin trays or cases in sets of five or six. The most useful colours and the 

' Exile V. Euler. A rule is an instrument with straight edges divided into inches and fractions of an inch (or into metric measurements). It is an instiument 
used for making linear measurements. The regula (ruler) of the ancient Romans was thus divided. A ruler or straight-edge is an instrument with straight edges 
(usually bevelled) for guiding a pencil, pen, or scriber in drawing straight lines. Thus, although a rule can be used as a ruler, to call the former a ruler would be 
a misnomer, one often used by non-technical vnriters. Bound desk-rulers are very convenient for drawing parallel lines for ledger and such like purposes by those 
accustomed to their use. 

- Not to be blotted out or effaced. 

' The quality of Indian inli difiers very much, but if good the stick will have a brownish glazed appearance at the end after being used. 



6 



INDUSTRIAL DRAWING AND GEOMETRY 



materials they are used to represent are given below. The first four will suffice if only ordinary metals are to be indicated ; the 
remaining ones are required when the other materials of construction, etc., are to be shown in colours. 



/ Prussian blue to represent wrought iron (and for dimension lines) 



Colours for ordinary metals 



Payne's grey 
Crimson lake 



Colours for other materials 
of construction, etc. 



Gamboge, or Indian yellow . . 
Prussian blue and crimson lake 

Yellow ochre 

Burnt sienna 

Sepia 

Light red 

Indigo lead 

Burnt umber 

French ultramarine .... 



cast iron 

centre and datum lines (and for dimension 

lines on tracings for blue prints) 
brass and gunmetal 
steel 
stone 
wood' 
leather 
brickwork 
lead 
packing 
water 



18. Saucers for mixing Colours. — The lid of the tin case of coloiirs that are sold for school use is stamped in the form of 
saucers for mixing the colours in; but the most useful saucers are the cabinet nests of white china, which are sold in sets 
of five and a cover. 

19. Brushes. — For colouring drawings you will require at least two brushes, the most suitable being a " middle swan " 
and a "small goose." These may be of camel hair,^ but preferably of red or brown sable hair. You will also require a camel-hair 




Middle 
swan. 



Fig. 14, — Brushes — sizes recommended for school use. 

water brush of about "large swan" size, for transferring water to the saucers, etc. These three brushes are shown full 
size in Fig. 14. 



' A camel-hair brush will not keep its point, nor spring back as well as a sable hair one does. But the former are much cheaper than sable, and answer well 
for rough work, but, not being so well made, the hairs often work out and adhere to the tinted surfaces. 



CHAPTER II 



HOW TO DRA.W STRAIGHT LINES AND SIMPLE FIGURES 

20. Introduction — It is a waste of valuable time for the beginner to attempt to draw even the simplest forms and objects without 
some previous practice in di'awing, in a workmanlike way, different kinds of lines, and a few representative symmetrical figures 
boimded by straight lines. You should carefully practise drawing the following progressive exercises, and after a few hours' work 
you should be able to draw simple plane figures neatly and with accuracy. Such operations seem so simple, but you cannot too 
soon find out that theory will not give precision in execution ; it is practice, guided by theory, with never-ceasing efforts to work 
with accuracy, that has gained for us as a nation supremacy in 
manufactm'es, and will gain for you the reputation of being a good 
draughtsman. 

21. Example. — Straight Lines drawn with the Assistance of the 
T-Square. — You should patiently practise with your pencil and 
T-square in the following way: — 

Commence by pinning the paper flat on the di'awing board; 
this can best be done by first pinning one corner until the under 
side of the pin- head is in close contact with the paper. Then press 
upon the paper near this pin, and move your hand diagonally across 
the sheet to the opposite corner, drawing the paper taut by the 
friction exerted. Hold this corner down by the thumb and fingers 
of your other hand, and insert a drawing pin as before described. 
Smooth the paper by hand from the centre to the other corners 
and pin them, and the sheet will be as flat as it is possible to have 

it by using pins only. The T-square can now be placed in position and held firmly by the left hand in such a way as to keep the 
stock in contact with the edge of the board, and the blade tight on the paper, as shown in Fig. 15. The pencil should be held 
between the first two fingers and thumb of the right hand, and kept in contact with the edge of the T-square, resting the third and 
fourth fingers on the square as the stroke is made. 

Y''ou should now aim at producing lines equal in thickness throughout their length, and, as the thickness and quality of a 
line depend upon the sharpness of the pencil and amount of unvarying pressure exerted upon it, you will understand that only 

7 



I^HVHB^I 




W^K^^^^^'^^B/l^^M 


ll^pPB^I 




::\%;^^^B 




LSii 






\mfML 







Pig. 15. — Showing how the T-square and pencil should be held. 



8 



INDUSTRIAL DRAWING AND GEOMETRY 



practice will enable you to draw tbem with certainty and facility. Each line should be drawn the full length of the T-square, and 
several of each kind should be drawn ; in fact, they should be drawn again and again till they can be freely produced at least equal 

in quality to those shown in the following figure (16), where it will be seen that A is a very fine line, 

suitable for centre and construction lines. This should be drawn with a very sharp chisel-pointed pencil, 
g and should be so fine that a light touch of the indiarubber will clean it out. At B is a line sensibly thicker 

than the previous one, and suitable for the finished lines of a very small drawing. C is thicker, and 
C suitable for ordinary drawing purposes. D is more suitable for working drawings of simple objects, drawn 

to a large scale ; and E is a suitable line for shade lines on drawings ; this line is best drawn with three 
^ strokes of the pencil, as the pressure necessary with a point thick enough to produce it with one stroke, 

p. would in most cases break the lead. The two outside ones should be sharp and distinct, and the distance 

between them decided by thickness ^ of the required line. In making the third stroke, the pencil should be 

turned sideways, so as to fill the space between the outer lines. 

The thickness and blackness of a line very much depend upon the pressure exerted on the pencil. 
21a. Defects in Lines. — The main defects in lines which should be avoided are : Varying thickness, caused by varying the amount 
of pressure exerted 'upon the pencil. Want of sharpness, the sides of the lines having a blurred appearance, caused by softness of lead 
or want of sharpness in the pencil. Uneven colour, due to Tinec[ual quality of the lead or paper, or uneven pressure upon the pencil. 



Pig. 16. — Thickness of 
lines. 






D 



Fig. 17. — Using set-square with downward stroke of pencil as at A, Pig. 18. 
22. Example. — Straight Lines, drawn with the Assistance of a Set-Sc[uare.- 



PiG. 18. — Diagram showing use of set-squares. 
-You should remember the instructions given for the 



'■ The ideal line of the geometrician has length only, without breadth ; but aU lines drawn by the draughtsman and executed in the arts have breadth as well 
as length. The term " line " is often used in referring" to things whose breadth or diameter is small compared with the length. Thus, ropemakers call cord, if small 
in diameter, line. And in writing and printing, the height of the letters being small compared to the length of the line, the term line is applied to the series of words 
running across the page. Again, in elevations or excavations of considerable length compared with their height or depth, the term line is oiten applied, as it is to 
the trenches and earthworks thrown up by the besiegers or besieged in military operations, also to rails upon which vehicles, etc., run. 



HOW TO DRAW STRAIGHT LINES AND SIMPLE FIGURES 



9 



previous example, and should now practise drawing similar lines with the assistance of one of your set-squares. The larger one 
had better he used, and the lines drawn its full length, at first to the right-hand side of the square as shown in Fig. 17 (and at A, 
Fig. 18), and afterwards to the left as shown in Fig. 19 (and at B Fig. 18) in the du-ection indicated by the arrows. It will be seen 
that the left hand in each case is firmly holding the set-square and T-square together and on to the board in such a way that the 
stock of the T-square is kept closely in contact with the edge of the board. The remarks upon the previous exercise respecting the 
quality of the lines apply equally to this one, and the necessity of practising the drawing of these lines from both sides of the set- 
square will be understood after your first attempts, as you will find that to steadily move your hand about with ease, in the required 
ways, needs considerable practice. 



^^^^^"^-^^Mnn 


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*N|1 'h^^KRb*^ 


^H 


w- ^9^ 


^r 


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BHB 


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B 



Fig. 20.— Dotted lines : different forms. 



E — 



Fig. 19. — Using set-square with upward stroke of pencil as at B, Fig. 18. 



Fig. 21. — Examples of dotted lines : 
difierent thicknesses. 



23. Dotted Lines.— Dotted lines are used on drawings either to indicate the line upon which a section has been taken or to 
mark the position of any existing part which is unseen ; for the former, dot-and-dash lines, as at A (Fig. 20), are used, whilst for the 
latter chain-dotted lines, B, should be used. In the former case. A, they look best when the dots areliqually spaced, and the short 
lines or dashes are equal m length, and about four or five times the lengths of the spaces ; and in the latter case, B, when of equal 
length and equally spaced, the lines being made three or four times the length of the spaces, as shown. Obviously, if the dashes 
are made shorter, they take a longer time to draw. The thickness of the lines, and the lengths of the spaces and dashes, should be 
regulated by the size of the drawing. A glance at some of the following lines, A to E (Fig. 21), will give you some idea of what is 
considered good proportion, showing how they should vary in form with the thickness ; and you should patiently practise drawing 
such lines until you can space them with a fair amount of neatness and facility. 

24. Rectang-les.— You should now be in a position to draw some simple figures. Having practised on lines drawn in the 
direction of the T-square, and at right angles to it, figures whose sides are made up of such lines should be easily drawn. So, by 



10 



INDUSTRIAL DRAWING AND GEOMETRY 



A C 



D B 



Fig. 22. — Construction of a rectangle. 
First step. 



AC D B 

Fig. 23. — Construction of a rectangle. 
Second step. 



Fig. 24. — Construction of a rectangle. 
The complete figure. 



carefully working the following progressive exercises, which are very fully described, you should make an important step in the 
practice of mechanical drawing. 

25. Example. — To draw a Rectangle whose Length (2") and Breadth (H") are given. — Draw, with the aid of the T-square, a 
very fine indefinite line, AB, about 2^' long (Fig. 22). With the aid of a rule and a pair of dividers prick off (Art. 14) the length CD 
equal to 2", and between these two points draw a good finished 
line as shown. Then, with the aid of a set-square, draw from 
C and D verj' fine distinct lines perpendicular to C and D, a 
little longer than the given breath (1^").^ Now, prick off as 
before the point E (Fig. 23) from C, making CE equal to 1^", 
the given breadth, and with the aid of the T-square, draw the 
finished line EF parallel^ 
to CD. 

The rectangle is com- 
pleted by re-drawing CE 
and DF (Fig. 24) with the 
aid of the set-square (being 

careful to regulate the thickness of the lines, so that they are the same throughout the figure), and removing with indiarubber the 
ends of the construction lines AC and DB, and those above E and F, leaving the rectangle completed as shown, care being taken 
not to remove the sharp corners formed by the intersection of the lines. 

Note. — You should always aim at constructing a figure by drawing the least number of lines possible ; in other words, a line should not be gone over twice, if 
once will suffice. As an illustration of this advice, with reference to the rectangle just drawn, many students would first have drawn the complete figure in fine 
lines, and then pencilled over each line to malse it of the required thickness. Such a practice usually produces a poor result, as it is difficult to exactly cover the 
the previous lines, and, further, it takes a longer time. 

26. Exercises upon the Use of Centre Lines. ^ — First Case. Figure Symmetrical about a Single Centre Line. — Whenever a figure 
has more than one line each side of its centre, and is symmetrical about that centre, it is best drawn by commencing witli the 
centre line. To illustrate this, let us proceed to draw the figure shown complete in the dimensioned drawing (Fig. 25). 

' The student, after a little practice, will be able to estimate these distances and lengths to vrithin a quarter of an inch, so that such lines need not be drawn 
much longer than their required length to minimize rubbing out, but in no case should they be drawn too short at first, as any attempt at joining a length on is 
usually noticeable, and should be avoided. 

= Parallel lines, at equal distances, are of frequent occurrence in the arts. When an engraver wishes to give us an idea of level and uniform surfaces, he 
represents the parts of them which are more or less in the shade by stronger or weaker lines, which are always parallel and at equal distances from one another. 
The ploughman forms his furrows in parallel lines ; and in music parallel lines at equal distances are used, the five lines are sometimes drawn at the same time, 
by means of a ruling pen with five points, at equal distances from one another. In a similar way the lines on drawings representing the wheel-tracks, or two rails, 
of rail-roads or tramways, are drawn. It has been remarked that the celebrated Cashmere shawls made by the Indians, which are remarkable for their fineness 
and beauty, cannot be compared to those made in Europe for uniformity of texture, as the Indians have not the same accurate instruments for preserving the 
parallelism and equal distances of the threads that the Europeans have. Thus the latter, by an approach to the precision of ideal geometry in the parallelism of 
straight lines, have obtained a superiority in an art practised for centuries and carried to great perfection in India. 

' Centre lines used in setting out work should be very fine continuous ones, undotted, as at A, Fig. 16 ; then any part of them can be used to measure to 
or from. 



HOW TO DRAAV STRAIGHT LINES AND SIMPLE FIGURES 



11 



Commence by drawing a very fine line AB (Fig. 26), with the aid of the T-square ; then with dividers prick off upon it two 
points C and D, 2" apart. Through these points, with the aid of a set-square, draw two fine indefinite lines EG and FH. Then, 
with the dividers, prict off on one of these lines, 
say from C, the points J and K (Fig. 27), the 
opening of the dividers being f", equal to a half 
of the breadth (Ij") of the given figure, and with 
the aid of the T-square draw through these points 
the finished full lines KM and JL. In a similar 
way mark off N and P from C, with the dividers 
open to j^", and through these points draw, in a 
similar way, the lines NO and PQ. 

The figure shoidd now be completed by 
going over the lines KJ and LM with the pencil, 
taking care to give the lines the same thickness 
and finish as the others, and the figure will be now complete as in Fig. 25. 

The projecting parts of the construction lines, should now be rubbed out with indiarubber, as in the previous exercise, the 

centre line AB being left projecting about I" beyond the figure on each side. 

Note. — The appearance and finish of the figure depend upon the lines being perfectly uniform in thickness and colour, and the student should constantly bear 
in mind the instructions previously given respecting the production of such lines. 



Fig. 25. — lieotangular figure, symmetrical about 
a centre line. The complete figure. 



H 

Pig. 26. — Rectangular figure. First step. 



c 

p 

K 
C 

Pig. 27.- 






.1 

-loi 








_ . 


-¥^ 


<- '"— » 




2 




















=H« 




1 





-Rectangular figure. Second step. 



Fig. 28. — Square figure. Use of two centre lines. 
The complete figure. 



Fig. 29. — Square figure. Construction lines. 



27. Seco7id Case. Figure Symmetrical about Two Centre Lines.— The complete figure. No. 28, consists of two concentric squares 
wliich are symmetrical about two centre lines, at right angles to each other. So, first draw any two indefinite centre lines AB and 



12 



INDUSTRIAL DRAWING AND GEOMETRY 



CD, perpendicular to one another (Fig. 29), and intersecting at E; then, with rule and dividers, prick off from E, along the 
centre lines distances EF, EG, EH, and EJ, equal to half the side of the outer square, viz. 1", and complete the square as in the 
previous case. The inner square should be drawn in the same way, the construction lines removed, and the required figure 
completed as shown in Fig. 28. 



-d- 






2i 



Fig. 30. — Complete figure symmetrical 
about two centre lines. 



Fig. 31.— First step. 



Fia. 32. — Second step. 



Fig. 83.— Third step. 



28. Another Case of a Figure Symmetrical about Two Centre Lines. — The figure to be drawn in this exercise consists of a 
rectangle, with a trapezoid at each end (Fig. 30). It will not be necessary to explain every step in the construction of this figure, 
as you should by this time be familiar with the method of working from centre lines, and might now attempt to draw the figure in 
what appears to you the best way, with a hint that the small ends ab and cd of the trapezoids should be drawn before the sloping 
sides. The Figures 31, 32, and 33 show the steps in the construction. These should speak for themselves now. Of course Fig. 30 
shows the complete figure. But you should not trouble about writing dimensions on your drawings yet. 

DRAWING EXERCISES. 

All the following exercises should be drawn full size, care being taken to make the lines sharp and d^ptinct, and the dimensions as accurate as possible ; these 
should be checked, where two or more occur in the same line, by scaling off the overall dimensions. Your teacher in awarding marks will take these points into 
consideration. 

1. Draw a straight line, using your T-square, and mark ofi from one end successive lengths of 2J", '&^{' , and 2]| '. Then measure the line formed by these 
three parts, compare this length with the calculated sum of the three quantities, and write down any error you may detect due to the faulty manipulation of your 
instruments. 

2. Draw 3 lines, A, B, C, any length, write on them their apparent or estimated lengths, then measure them with your rule, and tabulate the results as in 
table below. Note. — By this kind of practice you will educate your eye for measurements. 





Estimated L. 


Measured L. 


Error. 


A 








B 
C 















HOW TO DRAW STRAIGHT LINES AND SIMPLE FIGURES 



13 



3. Draw Fig. 34, commencing by drawing the bottom or base line. 

4. Two keel blocks are represented by Pig. 35 : commence by drawing the base line and 
vertical centre line. 

5. Pig. 36 shows the end view of a block of steiJS, whose risers and treads are repre- 
sented on the drawing by 3 cm. (3 centimetres). Commence by drawing the base and back 
lines. 

6. Commence Pig. 37 by drawing the horizontal centre line. 

7. The dovetail key, Pig. 38, is symmetrical about two centre lines : draw these first. 

8. Commence the concentric squares by drawing the two centre lines about which they 
arc concentric. 

9. The Maltese cross, Pig. 40, is symmetrical about two centre lines : draw these first. 

10. The panelled door represented by Pig. 41 is symmetrical about a vertical centre line, 
which may be drawn first. 



=-1* 


fnlcl 




- 3r- 


--^ 


">n|00 


'-1* 




Fig. 37. 



Pig. 38.— Dovetail 



Pig. 39.— Con- 
centric squares. 



•'""1 




1 


-JcN 



Fig. 34. 

I 



I I 

3|r 



h 



Fig. 35. — Keel blocks. 







T 


~'*nl^ 








1 


to 




■^ 


^8 


oj 


,1, 


^ 


1 


~--, 




5 
8 


.4:. 


S 
8 


./f.i 




II 


-^ 


1 






i, 


'•ol* 





Pig. 40.— Maltese 
cross. 



Pig. 41.— Panelled 
door. 



1 






3cm 


-t 




. J 


1 




- r 

r 


/2'cm - 


-- 



Pig. 36.— Block 
of steps. 



CHAPTER III 



MEASUREMENT AND CONSTRUCTION OF ANGLES, ETC. 

29. Introduction. — Before working- any problems relating to angles you should have clear ideas as to how angles are measured. An angle 
is selected as the unit, and the measure of any other angle is the number of units which it contains. Any angle might be taken for the unit— - 
as, for example, a i-ight angle — but it is obvious that a smaller' angle than a right angle would be more convenient. Accordingly a right angle is 
divided into 90 equal parts called degrees, the circle containing 360 ; ' and any angle may be estimated by ascertaining the number of degrees (or 
standard angles) contained by it. Thus, if CD (Fig. 42) be at right angles to AB, the angle BCD will contain 90°, and the arc AD wiU be a 
quarter of a circle; that is, CD wiU be inclined to AB 90°. Now, the angle ACD is equal to the angle BCD; therefore the angle ACD equals 





Pig. 42. — Construction of angles. 



Pig. 43. — Setting out an angle with the protractor. 



90°, and the angle between the arms of the straight ILae ACB wUl be 180°. Obviously, if we bisect the angle BCD in E, we get CE inclined 45° 
to CB. 

Other angles can be constructed with equal facility. Thus, if we take the angle ACE we get 90° + 45°, or an angle of 135° ; and by taking 
A as centre, AC as radius, and describing arc ACF, we divide the angle ACD in such a way that the angle ACF equals two-thirds of the angle 
ACD, that is, equals 60° ; and of course the remaining angle DCF must equal 30°. 

An angle of 15° is obtained by bisecting 30°, and 75° by adding 15° to 60°. 

30. To draw, -with the assistance of a Protractor, a Line making a given Angle (say 25°) with a given Line, and passing through 
a fixed Point in that Line. — Let AB (Pig. 43) be the given line, and C the fixed point in it. Then place the bottom edge of the protactor ^ 

' Thales, 640 B.C., first applied the circle to the measurement of angles. We still adhere to the practice of the ancient Egyptian astronomers, which was to 
divide the circle into 360 parts called degrees. Each of these degrees was divided into 60 parts called minutes ; these, again, into 60 parts called seconds, as we 
have seen. (We divide these seconds decimally, not into 60 parts called thirds, as the ancients did.) 

- These instruments are made in two forms as shown on the figure. The rectangular one being generally made of boxwood or ivory, whilst the semicircular 
one is invariably made either in metal, horn, or celluloid. 

14 



MEASUREMENT AND CONSTRUCTION OF ANGLES, ETC. 



15 



on AB, and move the mstrnnient along- it until tlie star or centre point, from which all the lines or degrees radiate is directly over the point C. 
Then mark the point, E, from the edge of the instrument, which corresponds to the reading 25° (of course there is such a point the other side 
of the protactor at E^), and through C and E draw the line CD, which wiU make an angle of 25° with AB as required. 

31. Use of the Set-Squares in Setting out Angles. — In ordinaiy mechanical drawing-, geometi-ical constructions and the protractor are never 
used for setting out angles if an easy manipulation of the set-squares will give them, and Figs. 44 to 49, which speak for themselves, show how such 
useful angles as 15°, 30°, 45°, 60°, 75°, 105°, 120°, 135°, and 150° can be drawn. 






Pig. 44. 



Pig. 45. 



Pig. 46. 



Pig. 47. 



Fig. 48. 



Pig. 49. 



32. Use of the Set-Squares in Trisecting and Bisecting Angles.— By using the 60° set-square, as shown in Pig. 50, a right angle, ABC, 
can be readily trisected. Whilst to bisect an angle, BAC, Pig. 51, with any suitable radius, and A as centre, cut the lines in B and C. Then apply 
either of the set-squares, as shown, and draw the lines CD and BD, intersecting in D. Join D to A, and this line bisects the angle. 

33. Copying Angles.— In copying an angle we may use either (a) a bevel, (6) an adjustable protractor, or (c) geometrical means. 

(a) A bevel is an insti-ument used by engineers, carpenters, etc., for drawing and transferring all kinds of angles. It is shown in Pig. 52, applied 
to an angle, BAC, and it consists of two parts or i-ulers, ED and EP, attached to and hinging on the same pin or pivot, P, in such a manner that angles 





Pig. 50.— Trisect- 
ing a right angle. 



Pig. 51. — Bisecting 
an angle. 



Pig. 52. — Bevel applied to angle. 



Pig. 53. — Copy of an angle 
by using a bevel. 



of every size may be f oi-med or copied by them ; the pin joint is a fairly stiff one, so that some little exertion is required to open or close the instrument 
for different angles ; then, once it has been set, as in Pig. 52, to the angle BAC, it can be used on a line GH, Pig. 53, and GJ can be drawn, making the 
angle HGJ the .same as the angle BAC. Or it may be used to ascertain whether the two angles are of the same magnitude either on a drawing or on a 
piece of work. 

(6) Angles may be copied by using some form of adjustable protractor, such as the useful little instrument invented by Prof. Low, and shown 
in Fig. 54. It consists of two parts, A and B, the former tongued on its circular edge, DEP, to snugly fit a corresponding groove in B (as .shown in the 



16 



INDUSTRIAL DRAWING AND GEOMETRY 



detail sketch at C, wliicli is a section at F). As will be seen, the circular edge of the part A is divided, so that the angle that MN makes with GH can 

he read off at the'division E. Thus this handy instrument combines the functions of a bevel and protractor for drawing purposes, and may in many 

cases be used as explained in connection with (a). 

(c) Angles may also be copied by geometrical 
means. A simple, very old expedient is to proceed as 
follows La copying the angle BAG (Fig-. 55). With any 
convenient I'adius, 'and centre A, describe an arc, DE, 
as shown. Then draw a straight line GH (Fig. 56), and 
with centre G and the same radius draw an arc KP. 
Next open the compasses to measure DE with the 
points, and with centre K draw an arc to cut FK in F. 
Then, obviously, KF equals ED. In fact, if the two 
chords of the arcs were drawn, AED and GKP would 
be two equal and similar isosceles triangles, with their 
angles at A and G equal, of course. 

34. The Fitter's Square. — The fitter's, or car- 
penter's square,' Fig. 57, is the instrument which is 

extensively used in the industrial arts in working on and arranging edges and planes which are to be at right angles (or square) to one another. The 

two working positions of the square are shown in the figure (57) at A and B. 

35. The Plumb ^ Rule, Fig. 58, is an instrument largely used in a number of trades, particularly in those relating to the construction of bxdldings, 

for testing the uprightness or verticaHty of walls, etc. A lead or brass bob is suspended from the upper part of a wood or steel straight-edge, as shown 




G H 

Pig. 54. — Low's adjustable 
protractor. 




E a 
-A given angle BAG. 




Pig. 



56. — Copy of the angle 
BAG in Pig. 55. 



M 



B 



Pig. 57. — The two ways of using 
a fitter's square. 




fl^g:^ 



Pig. 58.— Pig. 59 —Plumb line for 
Pliunb line. horizontal work. 



Fig. 60.— Spirit level. 



(with a centre line CD marked on it parallel to AB), by a piece of string, and its edge AB is offered to the wall ; if this is vertical the string or plumb 

line and CD wiU coincide, as the former always hangs in a vertical position when at rest. 

Fig, 59 shows a plumb line fitted to a large wood square, which allows of its being used for horizontal or level work. - 

36. The Spirit Level is another important instrument used for testing whether a plane is horizontal. It often consists of a block of hard wood 

of square section with a slightly cui-ved glass tube, full of alcohol or some limpid spirit (except a small bubble of air), fitted to its upper part, as shown 

1 Those used by engineers are wholly made of steel, whilst usually the carpenter's square has a steel blade fitted to a wood back or stock. 
^ The name is derived from the Latin word plumbum, lead. 



MEASUREMENT AND COiNSTRUCTION OF ANGLES, ETC. 



17 



iu Fig:. (iO. The tube is carefiiHj' einbedtled in plaster of Paris iu such a way that when the bottom of the block is resting- on a horizontal plane, the 
bubble of air settles in the middle of the tube. Should the body upon which the instrument rests be out of the horizontal, the bubble moves along the 
tube towards the hig^her end, and the more the tube is curved the shorter the distance the bubble moves, in other words, the less sensitive it is. 



EXERCISES. 

1. Set out the following angles, using your set-squares only : 75°, 105°, 120°, and 150°. 

2. Construct a triangle with sides i" , 5", and 6", and measure with your protractor its three angles. You, of course, will remember that in all triangles the 
sum of the angles equals 180°. So add the three angles together, and check your work. Repeat this by drawing other triangles of any shape ; you will soon find 
that great care must be taken in using the protractor to ensure accuracy. 

3. Set out a triangle with its sides 3", 4", and 5", and measure its angles. If you are acquainted with the 47th problem in the first book of Euclid, you will be 
able to satisfy yourself that one of the angles must be a right angle. 

4. Set out an angle of 30° with one of your set-squares, and then copy in the way explained in Art. 33(c), checking the copy by applying your set-squares 
to it. 

5. Set out at random any three angles. A, B, C, and write on each your estimate of its magnitude in degrees. Then measure them, and tabulate thus : — 



Angle. Estimated degrees. 


Measured degrees. 


Error. 


A 






" i 






C ! 

i 







Note.— Occasionally practise this and you will cultivate an eye for angles. In recent years teachers have found this type of experimental question of great 
educational value. 



CHAPTER IV 

CONSTRUCTION OF TRIANGLES 



37. Definitions, etc. Also refer to the definitions, etc., at the end of the book (p. 160). 

A triangle is a closed figure having- three sides and three angles. The sum of any two of its sides must be greater than the third (Euc. I. 20). 

The sum of the three angles of any triangle is 180° (two right angles) (Euc. I. 32). 

The perimeter of a triangle is the sum of its three sides. 

Similar triangles are those having egual angles, not necessarily equal sides. 

Apex or Vertex. — The angular point opposite the base of a triangle is called the apex or vertex. 

Median. — A lin e di'awn from a vertex to the middle point of the opposite side is called a median. 

The greater side of every triangle has the greater angle opposite it (Euc. I. 18). 

If a line which bisects the vertical angle A of any triangle ABC cut the base BC in D, the ratio of BD to DC is the same as the ratio of BA to AC 
(Euc. VI. 3). 

Congruent Triangles. — If the three sides of any triangle be bisected, and the points of bisection be joined, the four triangles formed will be similar 
and equal in all respects ; they are therefore caUed congruent triangles. 

We may now work a few representative problems on triangles. 

38. To construct a Triangle with Sides any Given Lengths, say, 2'5", 2-2", and 1*75". — First'draw one of the sides, AB 25" (Fig. 61), as 
base.^ Then with A as centre and radius of 2'2", describe an arc. With B as centre and radius of 1'75", describe another arc, cutting the former one 

in C Draw AC and BC, completing the required triangle. 

39. To construct an Equilateral Triangle with, say, 25" Sides. — The con- 
struction is similar to that in the previous problem. Draw AB 25" long (Fig, 62). With 

Q / / \ \ A as centre and AB as radius, describe an arc. With B as centre and with same radius, 

describe another arc cutting the former one in C. Complete the triangle by joining AC 
and BC (Euc. I. 1). 

Note. — An equilateral triangle has three equal sides, and therefore three equal angles. 
Then, as the three angles of any triangle together equal 180°, each angle wiU equal one-third of 
180°, equal 60°. So you may use your 60° set-square to draw the two sides. 

40. To construct a Triangle upon a given Sase (say 2V), the Angles being in 
a given Proportion (say 1:2: 3). — Draw the base AB 2^' long (Fig. 63), and with A 
as centre (any radius) describe a semicircle. Divide the semicircle with the di-^dders into 
six (1 -t- 2 -1- 3) equal parts, and number them as shown in the figure. Join 2 and 5 to 

ABC is the required triangle. 




Fig. 61. — Scalene triangle. Fig. 62. — Equilateral triangle, 
the centre A, and from B draw a line parallel to A5 till it intersects A2 in C 

' NoTB. — The line on which the triangle stands is usually called the base, but for geometrical purposes any side may be considered as such. 



18 



CONSTRUCTION OF TRIANGLES 



19 



Note. — The nnmber of parts into which the semicircle is divided is always the sum of the terms of the proportion. You wiU see that the angle CAB = 2A0, 
CBA = 5A6, and therefore ACB must equal 5A2, as the sum of the two angles 5A6 and GAO in the semicircle are equal to the sum of the two angles CAB and ABC 
in the triangle, and the sum of the three angles in each figure is equal to 180°. 

41. To construct a Triangle when the Length of a Side (say 2"), an Angle at that Side (say 65°), and the Perimeter ' (say 4J"} are 
given. — Draw the side AB 2" long (Fig. 64). At either end A set off AC at the given angle 65° and 2i" long (4i - 2) ; as the sum of the base and 
AC must equal the perimeter. Connect BC, and bisect it at D by a perpendicular cutting AC in E. 
Join BE, and ABE is the required triangle. I'NiC 

Note. — It should be noticed that EB equals in length EC ; therefore EB added to AE equals AC (Euc. I. 4). 

41a. To construct a Triangle when the Base (say IV'), Vertical Angle (say 35°), and Altitude 
(say Ij") are given. 

The working of this problem depends upon the fact that all angles in the same segment of a circle 
are equal.- 

Thus in the semicircle Fig. 65 the angles ABE and ADE are equal, and are right angles, as are all 
angles in a semicircle (Euc. III. 31). 

Again, in the segment Pig, 66 the angles FGI and FHI 
are equal, and, being in a segment greater than a semicircle, 
they are less than right angles (Euc. III. 31). 

The segment Fig. 67 is less than a semicircle, and the equal 
angles KLN and KMN are therefore greater than right angles 
(Euc. III. 31). 

To proceed with the problem. You will understand from 
the preceding remarks that the first thing to do is to di-aw a 
segrment with a chord li" long, such that an angle at the cir- 
cumference win equal 35°. This may be done thus. Draw 
AB ly' long (Fig. 68), and at its middle point F erect a 
perpendicular. At either end of AB draw a line incHned ^ 90° 
With radius EA and centre E, draw the arc ADCB 




Fig. 63. — Triangle, with angles in given 
proportion. 




35° = 55, intersecting the perpendicular at E. 
Draw DC parallel to AB and 1]" (the altitude) from it. 
Then join CA and CB, and ACB is the required ti-iangle (Euc. III. aS). 







Fig. 65. 



Fig. 66. 



Fig. 67. 



Fig. 68. 



' Sum of the three sides. 



- The angle at the centre is always twice the angle at the circumference ; thus PJI (Pig. 66) is twice FGI (Euc. III. 20). 
' This angle is always the complement of the vertical angle (Euc. III. 20). 



20 



INDUSTRIAL DRAWING AND GEOMETRY 



Notes. — 1. The dotted triangle ADB is the same shape and size as ACE, so that two triangles can be drawn to satisfy the problem.' Therefore this is called 
an ambiguous case. 

2. Of course the data could have been varied by giving an angle at the base instead of the altitude. 

3. You will remember that the altitude of a triangle is the perpendicular distance of vertex (corner opposite base) from base or base produced. 



42. To construet a Eight-angled Triangle, having given the Hypotenuse (say 2J") and One of the Acute Angles 
problem is best worked by putting the triangle in a semicircle whose diameter is equal to the hypotenuse. Proceed thus. Draw AB 
and describe a semicircle upon it. At either end B 

draw a Hne, making an angle of 25° with it. This 
line outs the semicircle in the point C. Join CA, and 
the triangle ABC is the required one, for as the angle 
ABC is in the semicircle, it is a right angle (Euc. III. 
31). Refer to Fig. 65. 

Note. — If the hypotenuse and a side had been given, 
a similar construction would obviously work the problem. 

43. To construct a Triangle whose Perimeter A 
C, 



(say 25°).— This 
21" long (Fig. 69), 






Pig. 69. 



Pig. 70. — Triangle, sides in given proportion. 



Pig. 71. 



shall be a given Length (say 5") and Sides in a given Proportion (say 2:3: 4).— Draw the line AB 5" long (Fig. 70). At A draw AG, 
making any angle with AB, and, with a distance between the legs of the compasses equal to about {. + 3 + 4 . °^ -^^j' '^^' al>out one-ninth of AB, 

as near as can be judged by the eye, set ofE nine of these distances along AC. Join the ninth (at C) to B, and di-aw through the second from A 
a line 2D, parallel to BC, and thi-ough the fifth from A a Hne 3E also parallel to BC. Then AD, DE, and EB are the three sides of^the required 
triangle, which may be constructed by drawing arcs from centres D and E, and radii DA and EB, respectively, intersecting in F. Join FD and FE, 
and the triangle is complete. 

44. To draw the Inscribed and Circumscribing Circles of any given Triangle.— Let ABC (Fig. 71) be the given ti-iangle. To inscribe the 
triangle with a circle, bisect any two of its angles (see Problem 32), such as ACB and CBA, by CO and BO, intersecting in O. Then with as 
centre, and radius equal to the perpendicular distance of O from either side, describe the circle. The centre of the circumscribing circle is 
f oimd by perpendicularly bisecting two sides of the triangle with the compasses, as shown ; thus MN bisects AB ; di-awing a similar line across CB, 
they intersect at P, which is equidistant from the corners A, B, and C, and is the centre of the required circle. 



CONSTRUCTION OF TRIANGLES 



21 



EXERCISES. 

Oral Questions. 

1. What is a triangle ? 

2. What is a scalene triangle ? 

3. What is the sum of the angles of any triangle in degrees ? 

4. Take your 60° set-s<iuaro and measure its sides. What do you find ? What is the ratio of the longest to the shortest side ? What is the name of the 
longest side '? 

5. Two angles of a triangle are found to measure 80° and 40°. What must the third angle measure, and why ? 

6. The angle at the apex of an isosceles triangle is found to be 49°. What are the angles at the base ? 

7. What is an obtuse-angled triangle ? 

8. Why are certain triangles called equilateral ? 

9. When are two triangles said to be congruent ? 

10. Your 45° set-square is the same shape as another boy's, although yours is smaller. Are the two figures they represent congruent ? if not, why ? 

Deawing Exeecises. 

11. Construct a triangle with sides 2-75", 2", and li". 

12. Construct an equilateral triangle with 5|" perimeter. 

13. Construct an isosceles triangle, base 2", and an angle at the base of 40°. 

14. Construct an isosceles triangle, 2" base, and vertical angle equal 40°. 

15. Draw a triangle whose perimeter equals 6|", the angles to be in the proportion 5:6:7. 

16. Construct a triangle with a perimeter of 7", the base being 2^", and an angle at the base 45°. 

17. Construct a triangle, perimeter 6", base 2". The length of the other two sides to be in the proportion of 2 : 1. 

18. Construct a triangle, its altitude 2J", an angle at the base 60°, and its perimeter 7-5". 

19. Construct a triangle, altitude 1'75", vertical angle 85°, base 1-5", and dxaw the circumscribing and inscribed circles. 

20. Draw a right-angled triangle, the vertical angle 85° and hypotenuse 3". 

21. The perimeter of a triangle whose sides are in the proportion of 9 : 7 : 4 is 7'5". Draw the triangle. 
Construct a triangle with a base of 2", one angle at the base of 40°, and a perimeter of 6". 
Construct a triangle whose sides are 10' 6", 10' 0", 16' 8". Scale J" = 1' 0". 

Construct a triangle with a base 2i", one angle at the base 50°, and the angle opposite the base 55°, 
Construct a triangle ABC, making the base AB = 2i", the side BC = 2", and the angle BAC = 45°. 

to satisfy the conditions of this problem. This is called the ambiguous case. 

26. Draw five diSerent-shaped triangles. Carefully measure the three angles ABC of each one, and tabulate these measurements as follows : — 



22. 
23. 
24. 
25. 



Note. — You will find that two triangles can be drawn 



No. of triangle. 


Angle A. 


Angle B. 


Angle C. 


A" + B° + C. 


Total error. 


1 










2 












Etc. 













V^ 



In filling up the last column you will have to satisfy yourself what the sum of the angles should be, and compare that with what your measurements make 
it in column 5. 



CHAPTER V 

SCALES, THEIR CONSTRUCTION AND USE 

45. Introduction. — If we wish to draw the elevation of a machine whose height^ is, say, 5', and length 12', upon a sheet of paper 
whose surface does not exceed two or three square feet in area, it is evident it would be impossible to make this drawing of the 
machine full size. Now, suppose we make a line 3" in length on the drawing represent a foot on the machine, then a line 
5" X 3" = 15" long would represent the height of the machine, and one 12" x 3", or 36" long its length; and we should speak of the 
scale as being one of 3" to the foot, and the fraction of the scale, as it is called (or representative fraction as it is sometimes called), 
would be — 

3 inches 3 1 
1 foot ^ 12 ~ 4 
If 



In the same way : 



1 

4 
i. 

2 

1 

n- 

4" 

41 



inch represented 1 foot, the scale would be ^^^ 

.i_ 

24 

L 
" 13 

1 
" 8 

1 
" 3 

3 
" "S 

X 



And if 1 inch represented 1 yard, the scale would be 
„ 1 „ 1 chain 



1 X 12 X 3 

1 



12 X 66 
1 millimetre represent 1 centimetre, scale would be -^q 
1 „ „ 1 decimetre 



1 

792 



1 metre 



100 

TOOO 



' The dimensions of machines, details, etc., are usually written in feet and inches ; the former, as we have seen, being indicated by the suflfix ', and the latter 
by the sufSx ". Thus, 5' reads 5 feet, and 5' 3|" reads 5 feet 3^ inches. Further, 0-783" reads decimal (or point) seven eight three of an inch, equal to f,"^ of an 
inch. When metric measurements are used, the following abbreviations, m., dm., cm., mm. respectively represent metres, decimetres, centimetres, and millimetres. 

Angles are measured in degrees, minutes, and seconds. Thus 45° reads 45 degrees, and 20°, 40', 50" reads, twenty degrees, forty minutes, fifty seconds. 

22 



SCALES, THEIR CONSTRUCTION AND USE , 23 

46. Drawing to Scale. — Of course, whenever practicable, the drawing is made the same size as the thing to be drawn ; the 
drawing is then spoken of as being Ml size. If the size of the object will not admit of its being drawn full size, then as large a. 
scale as is practicable should be selected. This applies more particularly to detail drawings, where every minute feature must be 
clearly shown. The great size of some work necessitates its being set out in detail on large specially prepared boards, whilst, on the 
other hand, the details of watches, clocks, and small instruments can only be satisfactorily shown when drawn larger than their true 
size. In every case, whatever scale is decided upon, care must be taken to draw all parts of the object to the same scale, and thus 
get an exact, although a reduced or enlarged, representation of it. Scales should always be constructed and drawn on important 
drawings, that are not fully dimensioned ; so that the various parts may, with the aid of a pair of dividers, be scaled off, and so that 
any alteration in size, due to the shrinking of the paper, will affect both scale and drawing alike. These scales must be constructed 
and divided with great care and accuracy. 

47. Engineer's Scales. — Although most of the drawings made by the beginner will be full-size or half-size, for which any 
ordinary rule can be used, yet after some practice he may be called upon to make them to a smaller scale, such as } or ^ full size, 
or even less, so that he wUl require an instrument with these scales marked on it. Such instruments are called Scales, or Drawing 
Scales, and they can be had made of various materials, such as cardboard, vulcanite, boxwood, ivory, and steel. The ordinary length 
is 12", and they are made with thin edges to enable a distance to be marked off from the scale to the drawing direct with pencil or 
pricker ; but a more accurate method is to take the distance off with dividers, as explained in Art. 14. 

Vulcanite scales should be avoided, as they expand and contract greatly with changes of temperature. On the whole, the best materials for them 
are boxwood and ivory. The scales are 3", 1^", 1", |", ^", |", 5", and 4" to the foot, or i, J, ^, ^, ^^, gW, jV, and t,V fuU size respectively. Two 
scales on each edge of the instrument. 

48. Construction of Scales. — Although you can never hope to make a scale with the accuracy that is possible when a dividing 
machine is used, that is to say, with the truth of a good ivory scale or steel rule, or even a boxwood scale, there is no reason why you 
should not, with care and a little practice, make scales accurate enough for some purposes ; indeed, you have seen that scales have 
to sometimes be set out on a drawing, and you should therefore make an effort to understand how this is done. You will see that 
the construction of scales is based on the equal division of lines, executed on the principle of similar triangles ; and the following 
example or two, worked in the form of problems, should put you in the way of making them yourself from suitable data, and better 
understanding their use. 

49. To divide a Line (say 3-5" long) into any number (say thirteen) of Equal Parts.— Let AB (Fig-. 72) be the line SS" long. At 
one of its ends, A. draw AC at any angle (an angle of 30" is a convenient one) ; now open the compasses so that the distance between the points is about 
one-thirteenth part of AB (by guessing, not by trial), and step along the line AC, making points H, I, J, etc., to C, the thirteenth point from A. Join 
CB, and through D, P, J, I, H, etc., draw lines parallel to CB, cutting AB in points in E, G, K, L, M, etc. These Unes wiU divide AB into thirteen 
equal parts. 

Note.— The truth of this and similar constructions can be proved thus. The triangle ADE is similar to the triangle ACB, because their angles are respectively 
equal. Therefore the side AD will be in the same proportion to AE as the side AC is to AB, and DC will be to EB in this same proportion ; but DC is one- 
thirteenth of AC, therefore EB must be one-thirteenth of AB. And so on for the other divisions (Euc. I. 26 and VI. 2, 4, and 10). 

50. To draw a Scale of ,V,, to read Feet and Inches, and to make it long enough to measure 4'. — A length of 1' will be represented by 
^'u of a foot, or by }j" = J of an inch on the scale, and the whole length of the scale will be 4 X 2" = 3". Draw a Kne AB (Fig. 73) 3" long, and 



INDUSTRIAL DRAWING AND GEOMETRY 



carefully divide it into four equal parts (Problem 49). Then each of these parts will represent 1' ; divide the iii-st division AC into twelve equal parts, 
.and these parts will represent inches. The scale may be finished by drawing the lines shown in the figure, and if they are figured in the way shown, 





Pig. 73. — Constriiction of a scale of j'j. 



Fig. 72. — Line divided into equal parts. 

which is the correct way, dimensions can be readily taken off with the dividers, by placing the points on the feet and inches m the required positions. 
Thus, to take off 2 9 , place one leg of the dividers on point 2', and the other on 9". The distance between the legs will represent 2' 9". 

51. To draw a Scale of li" to 1 yard, the Scale to show Peet, and be long enough to measure 3 yards.— The representative fraction of 



this scale is 



3 FEET 2 



3 X 12 8 X 3 X 12 32' 



and the length of the scale will be 3 x U' 



: 3|". 



b 



Pig. 74. 



Proceed as in the previous problem ; di-aw a line AB (Fig. 74) 

this length, and divide it into three equal parts ; then each of these parts wiU . 
represent 1 yard. Divide AC, the first of these divisions, into thi-ee equal 
parts, and each of these parts will equal 1'. The scale should be figured and 
completed as explained in the previous problem. 

52. To draw a Centimetre Scale or Rule, making it long enough to 



measure 13 Centimetres.—Refen-ing to the Meti-io Tables at the end of the book, we find that a centimeti-e = 0-394". Therefore 13 x 0394" = 5-122" 
- 5j^ veiy nearly. So draw a line AB SJ" long (Pig. 75), and divide it into 13 equal parts, as in Problem 49, and complete the scale by figuring, and 
drawing the parallels to AB, and the divisions. Of coui-se, a length AC (10 centimetres) equals 1 decimeti-e. 



/Dec. 






1 1 1 '1 1 1 1 1 1 


1 1 



O / 2 

k 



Fig. 75. — Centimetre scale or rule. 



/O 



12 



13 Cent. 



Diagonal Scales. 

■ 53. Diagonal scales are used when the divisions on an ordinary scale would become very minute. 

The principle of the scale can be explained by referring to Fig. 76. Let the problem be, to divide a distance DC by a diagonal line into anv number 
(say four) of equal parts. Draw hues CB and DA from the extremities of the given line DC, and perpendicular to it, making them any length ; with 



SCALES, THEIR CONSTRUCTION AND USE 



25 



the dividers prick off any four equal distances, CG, GF, FB and EB, along AB, then through GFE and B draw lines parallel to DC ; complete the 
figui-e by drawing the diagonal DB, cutting the Unes in HIJ. 

Now, the triangles CBD and FBI are similar, therefore ^ = ^^ = ^. That is to say, the distance FI is half the distance CD. And again, 

IF FB i . . ' 

ym — rTp ~ X- Therefore JE is half IF and a quarter DC. This simple erpedient is equivalent to dividing the given distance or line into four equal 

parts. The value of this principle can he realized when d C 

we notice that DC may be as small as we like. Thus 

make it jL", then EJ wiU equal } x j^;" = j\j". We 

may now proceed to construct a proper scale on these 

lines. 

54. To construct a Diagonal Scale to show 
Eighths and Sixty-fourths of an Inch. — Draw a 
line AB (Fig. 77) any number of inches in length 
(say 3), and divide it into inches as at E and F. Di^dde 
AE into eight equal parts ; each of these will be an 
eighth of an inch in length. Then at A and E draw 
perpendiculars AD and EO, and from A set off, along 
AD, "g* = 8 equal divisions (any convenient size), and A 
through each of these divisions draw a line parallel to 
AB. Then join 8 on AD to 7 on AE, and through 
6, 5, 4. 3, 2, 1 and E on AB di-aw lines parallel to 8, 7, and complete the scale as shown in the figure. 

The student will understand, after studying the previous figure, No. 76, that the divisions between the lines AD and D7 are J x J = 
etc. To take ofl any distance with 



\ 


H\ 


\ 


G 




l\ 


F 




J\ 


< 



Pig. 76. 




Diagonal scale. 



the dividers, say IJ-j" (this will equal 
Is + T!'4), place one leg of the dividers 
on Fl", where the horizontal line 7 
cuts it, and move the other leg till it 
is on the diagonal 1. Then the dis- 
tance between the legs will be the re- 
quired one, namely 1^". 

It ■w'ill be noticed that the product 
of the divisions in AE and AD (8x8 
= 64) equals the number of jiarts into 
which the distance AE has been divided 
by the scale. 

It foUows that, if the divisions 
had been 10 and 10, the line would 
have been divided into 10 x 10 = 100 
imrts, so that if a diagonal scale of 
yards is to be arranged to show read- 
ings of feet and inches, the divisions 
on the respective lines would be 3 and 12 



AO 



:-------ji-E 



2 H 3 F 

Fig. 78. — British and metric conversion scale. 



S inches 



55. To draw a British and Metric Conversion Scale. — In Problem 52 we found that 130 millimetres or 13 centimetres equal .5J". So take 



26 INDUSTRIAL DRAWING AND GEOMETRY 

a piece of squared paper with i" squares, and mark off a line AB 5^' long (Fig. 78). Draw AC perpendicular to AB, and 13 squares in height, and 
through C di-aw a parallel to AB, cutting a perpendicular at B in D, join AD, and figure the lines AB and AC as shown. Complete by drawing 
the perpendiculars throug'h the inch divisions on AB. 

tTse of Scale. — Example. — (a) Suppose you use a foot rule and find that the dimension of a body you are measuring is 3f ". Refer to your scale, 
and AF is this length, i-un your eye up the line FB, and you find that it cuts the diagonal AD in E, a point 1 of a square above the 9 centimetres Une, 
therefore your measurement is 9'2 centimetres. 

[h) Suppose that you are making a drawing in British measurements from a sketch with the dimensions in millimetres, and that you come to 
a dimension 60 mm., this is 6 centimeti-es ; running your eye along the horizontal Une through 6 on AC, you find it cuts the diagonal at Gr, which is on 
a vertical GH whose foot is 2f " from A, therefore your- equivalent length in inches is 2f ". 

EXERCISES. 

1. Draw a scale of 1 inch to the foot, making it long enough to measure 6 feet. Note. — You will draw two parallel lines about J" apart, making the upper 
one thin and the bottom one thick, as in Fig. 74. Set off the length 6", and divide it into six equal parts, and divide the first inch from the left-hand end into 
twelve equal parts, as in Pig. 72, and be careful to mark the divisions as shown. 

2. Make a scale of 1 inch to the yard, making it long enough to measure 6 yards. State what the representative fraction of this scale is. 

3. Draw a scale of 2 inches to the mile, and make it long enough to measure 4 miles, dividing the first mile into furlongs. 

4. Construct a scale of 1 inch to the chain, make it long enough to measure 8 chains, and divide the first chain into poles. Note. — The land chain is G6 feet 
in length, and there are four poles to the chain. 

5. Draw a scale of 2 inches to the pole, showing yards. 

6. Draw a diagonal scale to show tenths and hundredths of an inch, making it long enough to measure 6 inches. 

7. Draw a British and metric conversion scale. 



CHAPTER VI 

PROPORTION, ETC. 
Simple Problems in Proportion 

56. Introduction. — When foui- quantities — A, B, C, D — are proportionals, it is correct to say that A is to B as C is to D, and 
to write them thus, A : B : : C : D ; or thus, A : B = C : D. 

Thus it is easily seen that B bears the same proportion to A as D does to C, and therefore the ratio (when two quantities are 
compared with each other a ratio is formed) in each case is the same. And A and D multiplied togetlier will equal B and G 
multiplied together. This is called midtiplying extremes and means. Now, applying this to lines, we have A = 2", B = 3", 
= 4", and D = 6" ; and using these values for our proportion, we get 2:3: : 4 : 6, and the extremes and means multiplied give 
us 2 X 6 = 3 X 4. 

In geometry this is equivalent to saying that the rectangle made up of sides 2" and 6" is equal to* the rectangle with sides 
3" and 4". (See Chapter IX. on Areas.) 

Now, when the third term of a proportion is in the same ratio to the second as the second is to the first — thus, 3:6: : 12 : 24 
— the quantities are said to be in continued proportion. It is easily seen that 3 bears the same ratio to 6 that 6 does to 12 and 12 
to 24; and taking the continued proportion 3 : 6 : : 12 : 24, we have 3 X 12 = 6 X 6 ; that is, the first and third terms multiplied 
together equal the second term multiplied by itself (squared) ; and, further, 6 X 24 = 12 X 12 — that is, the second term multiplied 
by the fourth term equals the third term multiplied by itself. This is equivalent to saying that anij pair of alternate terms 
multiplied tor/ether will equal the intermediate term squared, or, in other words, the rectangle made up of sides equal to the alternate 
terms ^vill be equal to the square with sides equal to the intermediate term. The intermediate term is called a mean proportional 
between the other two. 

You would do well to thoroughly master the few simple principles of proportion, if you have not already done so, as many 
problems in mechanics and practical mathematics are easily solved by a simple geometrical application of these principles. 

In the previous chapter we have seen how problems on' the division of lines can easily be worked on the principle of similar 
triangles,^ and you had better again examine that before working the following interesting variation of it, which will conveniently 
lead up to the useful geometrical problems in proportion which follow it. 

57. To divide any given Line (say 275" long) into Three Parts so that the Parts will be in a given Proportion (say of 3 : 4 : 5). 
— Let AB (Fig. 79) be the line 2'75" long. At one of its ends. A, draw AC at any angle, and from A step along AC twelve 
' This is also the principle which governs the working of the proportional compasses, used for reducing the size of drawings, etc. 

27 



28 



INDUSTRIAL DRAWING AND GEOMETRY 



(3 + 4 + 5 = 12) equal parts. 



Join C (the twelfth part) to B, and from E, five parts from 0, draw EF parallel to CB, and from G, 

four parts from E, draw GH pa- 
i-allel to CB. Then the given line 




Pig. 79. — Line divided in given 
proportion. 





AB is divided at H and 
AH : HE : FB as 3 : 4 



F, so that 
5. 



Line divided in given ratio. 



A C D E B 

Fig. 81. — Division of a line. 



Direct Proportion 

58. To divide a Line AB (say 
3-5" long) in a Point C, so that 
BC : AC : : 4 : 5-5. Draw the line 
AB (Fig. 80), and at one of the 
ends, say B, draw BD at any angle, 
and from B step along BD nine 
and a half equal parts (5'5 + 4 



= 9-5). Mark this point D, and join AD, and from E, five and a half parts from D, draw EC parallel to AD. Then C is the point 
required. And BC will be in the same proportion to AC as 4 is to 5'5. 

59. To divide a line A2B2 Proportionally to any given Divided Line AB. — Draw the two lines parallel to one another as in 

Fig. 81. Join their ends AA2 and BB2, and produce the lines to meet in a point P. Through P draw lines to CDE, cutting 

A2B2 in C2, Da, and E2. You will then have divided the line A2B2 in a similar way, or proportionally to the given line AB. 

Note. — When the given lines are nearly equal in length, you must place them as near together as possible to keep the point P within the edges of your 
drawing paper. 

60. To divide the Space contained between Two Parallel Lines AB and CD into Equal Parts (say 6) by Lines Parallel to them. — 
Draw CK (Fig. 82) any perpendicular to them, and set off 1, 2, 3, 4, 5, 6 equal spaces; then with centre C, and C6 radius 

describe an arc 6 J, cutting AB in 
J. Join C to J, and with centre 
C describe the arcs IE, 2F, 3G, 
etc., and through the points E, F, 
G, etc., draw lines parallel to AB. 
These divide the space as required. 



Note. — Suppose AG represented the 
height of a wall of sis courses of brick- 
work, you have a means of finding the 
divisions representing the courses, or 
layers of bricks ; but of course the result 
is only satisfactory when the drawing is 
accurate. 





Fig. 82. — Division of a space. 



Fig. 83. — A third proportional to two given lines. 



61. To find a Third Proportional to Two given Lines, A and B.— Let A and B (Fig. 83) be the given lines. Then a line 



PROPORTION, ETC. 



29 



X is requii-ed, such that A : B : : B : X. Draw any indcfiuite Hue, IP. At 1 draw the line 10, making any angle with IP. 
From 1 on the line IP cut off 1 2 equal to A, and from 1 on the line 10 cut off 1 3 equal to B. Then, using 1 as 
centre, describe an arc 3 4, cutting IP in 4. From 4 draw a line 4 5 parallel to 2 3. Then the line 1 5 is the required third 
praportional X. 

Note. — If we state the relationship of the four quantities in the form of an equation, namely, AX = '' ,. B, we have the product of A and X equal to B=. 
That is to say, the rectangle made up of sides A and X equals the square on B. Further, B is the mean proportional of A and X. Eeter to Problem 64. 

62. To find a Foui-th Proportional to Three given Lines, A, B, and C. — In this pjcohlem the fourth term, X, in the proportion 
A : B : : C : X, is to be found. 

Take, as in the previous problem, any indefinite line, IP (Fig. 84), and at 1 draw 10 at any angle with IP. From 1 on 
IP cut off a length 1 2 equal to A, and on 10 from 1 cut off 1 3 equal to B. Then from 1 on IP cut off 1 4 equal to the 
third line C. Connect 2 and 3, and through 4 draw 4 5 parallel to 2 3. Then 1 5 is X, the required /o!(?-(;/i proportional. 





__A/_C 



30' 



Pig. 84. — Fourth proportional to three given lines. '""^ Fig. 85. — Findirig the height of a tower. 

63. The Shadow cast on the Groimd by a Vertical Rod 6' high is 7' 6" in length. Find the Height of a Tower whose Shadow 
is 30' long. Scale ^^" = 1 foot. — Obviously, the required jieight can be found by proportion. Again making use of similar 
triangles, draw a straight line AB (Fig. 85) to represent the shadow of the tower. This will be, to the given scale, 30 x y,/' or 
3". Along AB from A mark off AC equal to the 7' 6", the length of the rod's shadow, and at C erect a perpendicular to 
AB, making its length CD equal 6', the height of the rod. Next, carefully draw through A and D a line cutting a perpendicular 
to AB at B in E. Then the length of BE will represent the height of the tower. 

Note. — Of course the figure represents a side view of the rod and tower. It will at once be seen that the solution of this problem is an application of the 

AB RF 
well-known property of similar triangles, namely, that their similar sides are proportional (Euc. I. 26 and VI. 4), as explained in Problem 43. Thus 

or the required height is a fourth proportional to the three other quantities. 



AG ~ CD' 



30 



INDUSTRIAL DRAWING AND GEOMETRY 



64. To find the Mean Proportional C of Two given Lines A and B. — Draw any indefinite line OP (Fig. 86), and along it mark off the 
distances^ 01 equal to A, and 1 2 equal to B. Bise'^t tlie line 02. Let 4 be the point of bisection ; then, with 4 as centre and radius 40, describe a 
semicircle, and at 1 raise a perpendicular to cut the semicircle 

\in 3.-- "The Hne 1 3 is the required mean proportional C (Euc. _ _5 

<yi."13). ' ' ""' 

N^KES. — 1. The mean proportional squared 'i^' nal to the 
product of the two given lines ; that is, A x B = C, Oi A ; C : : 
C : B. As you will remember, the product of the extremes (first 
and fourth terms) of any proportion is equal to the product of the 
means (second and third terms). 

2. The mean proportional C is called the geometrical mean of 
A and B. The area of a square on C is equal to the area of the 
rectangle whose sides are A and B, as we have seen, and as shown 
in Fig. 87. 

This must not be confused with the arithmetrical mean of A 
and B, which is half their sum. That is, arithmetrical mean 
_ A + B 




Pig. 86. — Mean proportional. 



Pig. 87. — Square and rectangle equal in area. 



3. Yoh should study the relation the above problem bears to Problem 61. 

4. When B equals unity, C wiU equal the ,,/A, and the length of the line A represents the area of the rectangle formed by A and B. 

5. The line 1 5 is the mean proportional of 01 and IP, or of A and D, and the square of the two mean proportionals 1 3, and 1 5, are in the same ratio as 
B is to D. 

(13)- _ A X B _ B 
(1 5f ~ A X D ~ D 
You wiU easily understand that if 1 3 and 1 5 be two corresponding sides of similar figures, the lines B and D wiU represent the ratio of their areas, 
fore this construction is very useful in handling such problems as No. 115, when the ratio of the areas is given. 



For 



There - 



EXERCISES. 

Obal Exebcises. 

1. If four quantities are proportionals, what relationship must exist between the first and second, and third and fourth ? 

2. When are four quantities said to be in continued proportion ? 

3. What relationship must exist between three terms of a proportion if the intermediate one is a mean proportional of the other two ; 



Dbawing Exbboisbs. 

4. A line 6J" in length is the perimeter of a triangle whose sides are in the proportion of 2 ; 3 : 4. Divide the line to find the lengths of the sides of the 
figure. Measure these lengths with your rule and write them down. 

5. Two boys wallv towards each other from the ends of a straight road a mile in length. One walks at the rate of 2J miles per hour, and the other at 3^ miles 
per hour. Draw a line 6'^in length to represent the mile, and geometrically find the point in the line where the boys would meet. Measure its distances from 
theeiids. 

6. A line 5'4" in length represents, to a scale of one-twelfth, the height of a bookcase, and the positions of the shelves are indicated by divisions 1-7", 3-2", and 



PROPORTION, ETC. 

■ ?%- 

4'4" from one end of the line. Another line, 4'5" in length, represents the height of a second bookcase whose shelves are to be arranged at heights proportional 
the first one. Geometrically find the positions of the shelves of the second bookcase, and carefuUy measure and write down the distances bs,tween them.' "^ 

7. One side of a rectangle whose area is equal to that of a square of li" side is 1". Find by the method of proportion the length of the- other side. 

8. The sides of a rectangle are 2'75" and 3'6", and the long side is lengthened by 1". Find how much the other side must be lengthened i£ the si 
remain in the same proportion. " — , 

9. The shadow cast on the ground by a lamp-post, whose height ia 9', is 10', and the shadow oast by a telegraph pole is 23' 6"? Geometrically find the 
of the pole. . ,. -^-'""''^^ 

10. Find the mean proportional of two lines whose lengths are 2'8" and 1-6", and measure its length. ' / 



sides are 




.<cr' 



V- 



CHAPTER VII 



CIRCLES, ARCS, AND LINES 



65. Introduction. — As ordinary mechanical drawings mainly consist of combinations of circles, arcs, and lines, the art of correctly and neatly drawing 
a few of them in -various positions in relation one to the others, representing typical cases, should be cultivated by the beginner; for if such lines are 
faulty in form and finish, or do not satisfy the geometrical conditions of proper contact, they spoil the appearance and detract from the value of any 
drawing upon which they appear. 

A few of the more important definitions and problems relating to circles and arcs are given here to help you, but for more complete information on 
these matters refer to definitions, etc., at end of the book ; you may also refer to the author's " Geometrical Drawing," p. 61, etc. 
66. Definitions. — The radius of a circle is a straight line drawn from the centre to its circumference. 
A diameter of a circle is a straight line passing through its centre, and terminated at both ends by the circumference. 
An arc of a circle is any part of the circumference. 
A chord is a straight line joining the extremities of an arc. 
A segment is any part of a circle bounded by an arc and its chord. 
A semicircle is half a circle, or a segment cut ofE by a diameter. 

A sector is any part of a circle bounded by an arc and two radii drawn to its extremities. 
A quadrant, or quarter of a circle, is a sector having a quarter of a circumference for its arc, and the two radii perpendicular to each other. 

A sextant, or sixth of a circle, is a sector 
having a sixth of the circumference for its arc, 
and the two radii making an angle of 60° with 
each other. 

An octant, or eighth of a circle, is a sector 
having an eighth of the circumference for its 
arc, and the two radii making an angle of 45° 
with each other. 

A tangent is any line perpendicular to a 
radius at its extremity in the circle. A tangent 
touches the circle in a point, as at P, Fig. 88 
(which is called the point of contact), where the 
line AB touches the circle, and it is perpendicular 
to the radius OP. 

Point of Contact. — Where two circles 
touch one another, they do so in a point only, 
called the point of contact, and the straight line 
which joins their centres passes through this point. Thus, Fig. 89 shows two circles, A and B, touching one another in the point P, which is the point 




Pig. 



38. — Circle and 
tangent. 



Fig. 89. — Point of contact of 
two circles. 



Fig. 90. - 



-An arc described through three 
points. 



CIRCLES, ARCS, AND LINES 



33 



of contact. It is only when tliis condition is satisfied that a part of one circle can be made to flow into a part of the other ; the thick line in the figure 
shows how this condition must be satisfied. 

To enable you to correctly treat cases where circles are in contact with one another and with straight lines, you should oaref idly study and work 
the following problems before attempting the exercises at the end of the chapter. 



Some Important Cases worked as Problems. 

67. To describe a Circular Are through Three given Points.— Let A, B, C (Fig. 90) be the given points. Join AB and 
perpendicular bisectors GrF, DF, intersecting in F. Then, with F as centre and radius FB, describe the required arc ABC. 

68. To find the Centre and Radius of a given Arc or Circle. — Let A, B, and C (Fig. 90), be any points in the given arc or 
AB and BC. and perpendicularly bisect the lines AB and , 

BC by the Hnes G-F and DF, which intersect in F the • 

requu-ed centre. 

In the case of a circle, if one of the bisectors GF or 
DF be produced both ways it cuts the circle in the ex- 
tremities of a diameter, and by bisecting the diameter the 
centre can be found. 

69. To draw a Tangent to a Circle through a , 
fixed Point in its Circumference. — Let B (Fig. 91) be / 
the fixed point in the circle. Join B to the centre A, and I 
thi'ough B draw CD perpendicular to AB. Then CD is „ 
the tangent required. 

70. To draw a Tangent to a Circle through a 
fixed Point without it. — Let the circle in Fig. 91 be the 
given one, and P the j)oint. Join the centre A with P, 
the fixed point without the circle, and bisect AP in B. 
With E as centre, radius EA, describe the semicircle 
AFP, cutting the given circle in F. Join PF. Then PF is the requii-ed tangent. 



BC, and draw the 
circle. Then join 




C 13 

Fig. 91. — Tangents to a circle. 



F' A 

Fig. 92. — Circle touching two 



F 
given lines. 



It is evident that in a similar way a tangent the other side of PA could have been drawn. 



Notes. — 1. You will notice that if P be joined with A, the angle PFA wUl be a right angle, being an angle in a semicircle. (Euc, III. 31.) And PA will be 
a normal to the tangent at P. 

2. The Euclidean geometry does not allow a tangent from a fixed point to a given circle to be drawn without first finding the point of contact as above, and 
the same remarks apply to the case of a common tangent to two circles ; but for practical drawing purposes a tangent may be drawn from an external point to a 
circle, or a common tangent to two circles directly by carefully adjusting the straight-edge ; and should the actual point of contact be required, a perpendicular to 
the tangent from the centre fixes it. 

70.1. To inscribe in a given Angle a Circle of given Radius (say 15"). — Let EAF (Fig. 92) be the given angle. Bisect the angle by the 
line AB, and draw CD jjarallel to AF and 1'.5" from it, intei-secting AB in C. With C as centre, radius 1'5", draw the circle touching the sides of the 
angle in E and F. The exact points of contact can be found by drawing from C the lines CE and CF perpendicular to AE and AF respectively.' 

The dotted lines refer to a case when the angle is obtuse, and the same letters apply. 

Notes. — 1. This is a problem often met with in mechanical drawing, when two lines are to be connected by an arc of a circle of given radius. 

' AE and AP are two tangents to the circle from A, and they are equal to one another (Euc. III. 17), 



34. 



INDUSTRIAL DRAWING AND GEOMETRY 



2. For most practical purposes a common tangent to two given circles (such as E'E, Pig. 92) can be drawn with a sufficient degree of accuracy by ofiering 
the edge of a square to the two circles and drawing a line to touch them, the points of contact being found by drawing perpendiculars from the centres to the 
tangent as shown. 

70b. To describe a Circle of given Eadius to touch a given Line and a given Circle. — From C (Fig. 93), tte centre of the given circle, 
draw any line CE, cutting the circle in P ; from F mark off FE, equal to the given radius, and with centre C, radius CE, describe the arc ED. -At 

any point H in AB di-aw GH equal to the given radius 
EF, and periiendicular to AB. Through G- draw GD 
parallel to AB, and cutting the arc DE in D. Then, with 
D as centre, radius EP, describe the required circle. 
Note. — D is equidistant from line and circle. 

71. To describe an Are of a Circle of given 
Eadius (say 1") to touch, a given Arc and a given 
Straight Line. — This is an obvious variant of the pre- 
ceding problem. AB (Fig. 94) is the given line, and 
G the centre of the given arc CD. Draw GO, any line 
passing throug'h the centre of the arc G, and cutting 
it in C. Mark off CE equal to the given radius of 1", 
and with centre G, radius GE, describe the arc EF, and 
draw F'F parallel to AB and 1" from it, intersecting 
EF in F, which is the centre of the required arc. 
Draw through G and F the line GK, cutting the circle in K. Then 




Fig. 93. 



-Circle of given size touching a fixed 
line and circle. 



straight line and arc. 



With F as centre, radius FH, a perpendicular to AB, describe the required arc. 
the points of contact are H and K. 

Q If the arc were to touch the given circle externally, F' would be its centre, and I and J 

its points of contact. The working is similar, and can be easily followed on the fig-ure. 

Note. — This problem occurs when a wheel with arms is drawn. HB is then the side of an arm, 
and the arc CK a part of the rim. 

72. To describe an Arc of a Circle to touch a given Line in a fixed Point and also a 
given Arc. — Let AB (Fig. 95) be the given line, E the fixed point in it, and HDL the given 
arc, whose centre is K. Draw from E a line CE perpendicular to AB, and through K draw 
KL parallel to CE, and cutting- the given arc in L. Join LE and produce it to cut the given 
arc in D. Join D to K, cutting CE in F. Then with centre F and radius FD describe the 
required arc DE, which satisfies the conditions of the problem. 

73. To dra-w a Circle to touch Three given Straight Lines.— Let the given lines be 
AB, AC, and CD (Fig. 96), iatersecting in A and C. Bisect the angle BAC by the line AE. 
The centre of the requii-ed circle must be somewhere in this line. Bisect the angle ACD by 
the Une CF ; the centre must also be somewhere in CF. Therefore it is in G, the intersection 
of AE and CF. From G di-aw GH perpendicular to AB and cuttiag it in H. "With centre G, 
i-adius GH, describe thQ required circle or arc.^ Then perpendiculars from G, such as GH, give 
the points of contact. 




Fig. 95. — Arc touching a 
given arc and a Une in a 



A H B 

Fig. 96. — Circle touching 
three straight lines. 



fixed point. 
Note. — This problem sometimes occurs when a small bevel wheel is drawn, and AB is part of the boss or hub, and CD is the back of the rim. 

' Three other circles can be drawn to touch the given lines, and one of them will obviously be contained by the triangle made by producing DO and BA till 
they meet. 



CIRCLES, ARCS, AND LINES 



74, To describe Two Ares to meet eaeli other in the Line of their Centres and to touch Two given Lines at Fixed Points in thera, the 
Radius of the Lesser Arc being also given. — Let AB and CD (Fig-. 97) be the g-iven lines, and E and F the fixed points. Draw EG perpendicular 
to AB and equal to the radius of the lesser arc, and from F di-aw FH perpendicular to CD and equal to EG, Join G to H 
and bisect it by the perpendicular KN cutting FH produced in N. Join NG and produce it. Then with N as centre, and radius 
NF, describe an arc meeting- NG produced in L. With G as centre, radius GL or GE, describe the arc EL, Then the arcs FL 
and LE ai-e tangent to the given lines CD and AB respectively in the points P and E, as required, and the two arcs meet at L in 
the line NG of their centres, 

75, Interesting cases of Arcs flo-svitig into one another. — The few examples of proper tangential contact of line and 
circle, and of cii-cles of different curvature, shown in Figs. 98 to 104, should now speak for themselves, and you should be able 
to di-aw any of these in such a way that the geometrical conditions of proper meeting or contact are conformed to. You will 
not f-ail to notice that in such cases as Fig-. 99 the centi-es A, B, and C of the three arcs are all on the same line MN, And that 
the points D, E, and F (Fig. 101), each common to two arcs, are in the lines of centres BC, CA, and AB respectively. 

If von are fond of devising geometrical patterns for decorative -work you will soon begin to find out endless arrange- 
ments of circles and lines that will have a pleasing effect if they are properly drawn, 

76, Hints on working the Exercises.— Having studied the preceding- problems, the student should be able to work the 
foUowin"- exercises without further help. They should be carefully constructed from the dimensions shown, and not merely 
copied, "Having pinned down a sheet of paper, the T and set-squares should be carefully dusted, and the pencils and lead of the 
pencil bows to be used shai-pened, and the latter adjusted so that the pencil and steel point are of equal length ; the exercises can then be proceeded with. 




IMPORTANT TANGENTIAL ARCS. 







Pig. 100. 



Fia, 101, 





FiQ. 98. 



Fig. 102. 



Pig. 103. 



Pig. 104, 



36 



INDUSTRIAL DRAWING AND GEOMETRY 



EXERCISES. 

The diagram relating to each of the following problems should be drawn fuU size before attempting the solution. 

1. Assume any point P in the given circle (Pig. 105), and draw a tangent at the point. 

2. Through the fixed point P (Pig. 106), draw a tangent to the given circle. 

3. In the angles ABD and CBD (Fig. 107) inscribe arcs of 1^" radius. 

4. Describe a circle of If" diameter to touch both the line AB (Pig. 108) and the given circle. 

5. Describe a circle touching the three given lines, BA, AG, and CD (Pig. 109), and mark the points of contact. 

6. Describe a 2}" circle to touch both the given circles (Pig. 110). 

Y. Describe an arc to touch the line AB (Pig. Ill) in the point P and to flow into the given arc DE whose radius is 2" and whose centre is C. 
8. Describe two arcs to meet each other in the line of their centres. One of them to touch the line AB (Pig. 112) at E, its centre being P, and the other to 
touch the line AC in the point D. 




Pig. 105. 



Pig. 106. 



Pig. 107. 



Pig. 108. 



Pig. 109. 



Pig. 110. 



Pig. 111. 



<^ — £. 1 — t- 



Pig. 112. 



CHAPTER VllI 



USE OF SQUARED OR SECTIONAL PAPER 

77. Introduction. — Bel'ore comiueiiciug to study the interesting problems in this chapter, you should read the introduction to the 
chapter on Areas : you will then see how squared or sectional paper may be used in connection with areas of plane figures, and after 
you have examined the applications we shall deal with directly, you will understand what great use can be made of it in many 
branches of practical work ; further, if you are fond of thinking things out for yourself, you will 
soon be using squared paper in a number of interesting ways. 

78. Different Kinds of Squared Paper. — Squared paper is made with a great variety of rulings, 
and you should remember that the extra quality papers have the lines on them lithographed ; 
they are then more mathematically accurate than papers that are merely ruled. Fig. 113 shows 
a sample of squared paper with J," Euling, and the inch lines thick. Among the other rulings 
commonly in use may be mentioned : }" Euling, i" lines thick ; ^" Euling, 1" lines thick ; 
]J„" Euling, ^" lines thick ; 2 mm. Euling, cm. lines thick ; Millimetre Euling, cm. lines thick ; 
and y Euling. 

The sheets can be had either ruled on both sides or on one side only, and they are commonly 
made 23" x 18" and 16;^" X 13;}". Squared paper can also be had in rolls of 10 or 50 yards by 
26" in breadth. Square tracing papers can also be had with the same rulings. 

79. Use of Squared Paper. — The following few articles will give you a good idea of the 
usefulness of squared paper in working simple problems in co-ordinate geometry, and in graphi- 
cally solving algebraic equations. 

80. Position of a Point in a Plane. — In fixing the position of a point in a plane, say the 
plane of the paper, it is convenient to use two lines, OY and OX, Fig. 114, intersecting at right 
angles as axes or lines of reference; the point where they intersect or meet is called the origin. Let us suppose that P is a point 
12 units from^ OY, and 6 units from OX, the unit in this case being J". Then the position of the point is referred to as "the 
point (.^', y)," or as " the point (12, 6)." And the lines P6 and P12 are called the rectangular co-ordinates of the point P. 

81. Position of a Straight Line in a Plane. — Obviously, the position of a straig-ht line in a plane can be fixed by statuig the positions of its ends, 

' It is nsual to state the distance of a point from OY first. 
37 



Fig. 113. — Squared paper J" ruling, 1" 
lines thick. 



38 



INDUSTRIAL DRAWING AND GEOMETRY 



or, indeed, tlie iDositions of any two points in the line, because only one straight line can be drawn tlirongh two points. In Fig. 115, the line PP., passes 
thi-ough the origin, and if you di-aw through P, lines PN and PM (called co-ordinates) perpendicular to the axes, and di-aw similar lines tlu-ough Pa, 

'"2 



as shown,^ you wiU find that ■ 



Wr 



3, jj, „ y^ ^ But, of course, for the same sti-aight line the ratio — is constant, and may be written k. "We may 
then write the above equation y = kx. 

And in this form it is known as the equation of a straight line, which passes through the origin. Pui-ther, a straight line passing through the 
origin is referred to as a graph, of the equation y = lex. 



























~i 


1 


~" 


~~ 


■" 




^ 


6 


























1 n 




































>\ 





















































































































































































_^ 




L 




L 


u 


Llj 


J 


^ 


_ 




« 


_ 


_ 




^ 



X 

Pig. 114. — Position of a point in a plane. 




m^ 


nr> 




^^'r 




^' 




^^ 




^■' 




^'^ 


^'^ 




^^ 




_ 




_ ^^ 




del 


X 




20 



Fig. 115. — Position of line in a plane. 



Fig. 116.— Graph oiy = Ix. 



82. To dra'W a Graph. — Let y = hx. This being of the form y = hx, you now know that the graj)h is a straight line which must pass through, 
the origin. So the first thing to do is to substitute some convenient value for x, say 20 (as large as your paper will take), and calculate the value of y, 
namely, 1/ = i x 20 = 10. "With these values as co-ordinates, a second point in the gTaj)h on your squared jiaper is found, which, joined to the first, 
the origin, g'ives the required graph, as shown in Fig. 116. 

83. Straight Lines that do not pass through the Origin, and their Equations. — Let us assume that we are to draw the graph of the equation 
X = oy — 20. This is called an equation of the first degree ; that is to say, no factor in the equation is a squared quantity or higher order. Tou can 
easily jwove, as in Article 81, that all equations of the first degree represent straight lines, and, conversely, that all straig-ht lines can be represented by 
equations of the first degree. The graph is easily drawn if you first find two or more values - of x for corresponding values of y, namely, 0, 10, 20, 30, 
and 40. 

Thus, let 2/ = 0, then a = (3 x 0) - 20 = - 2(1 

„ 2/ = 10 „ » = (3 X 10) - 20 = 10 

2/ = 20 „ K = (3 X 20) - 20 = 40 

1/ = 30 „ IB = (3 X 30) - 20 = 70 

y = 40 „ iB = (3 X 40) - 20 = 100 

and so on. 

' You can do this by actual measurement, but of course you wiU remember that all such triangles as 0PM on the figure will be similar, and therefore their 
bases and perpendiculars will be in the same ratio. 

"■ Of course, points in the curves can be determined by giving arbitrary values to either x or y. The curve in this case being a straight line, of course two 
points only in it will fix its position in relation to the axes OY, OX, as you have seen ; but it will serve as an interesting and useful exercise to find two or three 
additional ones. 

We are here treating x as an independent variable, as it is caUed, y being the dependent variable ; but you need not trouble about these terms at present, unless 
they have been previously explained to you. 



USE OF SQUARED OR SECTIONAL PAPER 39 

Now, to plot tlie line, you may commence by taking- the first value of y. You find it is 0, which means that it must be somewhere on the line OX 
(Fig. 117), but the corresponding' value of x is —20, which means tliat it must also be twenty units from the axis OT, but to the left of O, as it is a 
minus quantity. (The positive values of x and y are measured to the right and upwards, while the negative values are measured to the left and 
dowuM-ards respectively.) The point P satisfies these two conditions, and obviously the points P^, P,, Pj, and P, equally satisfy the above values of their 
co-ordinates .<• and y. and all these poiuts, you will find, are on the straight Une PPj. 

84. Position of a Eectilinear Figure in a Plane.' — If you understand how the position of a point is fixed in a plane in relation to the two axes 
or lines of reference, as explauied in Article 80, you will see that the three poiuts, A, B, and C (Fig. 118) (the corners of a triangle), are fixed by their 
co-ordinates, or each one by two numbers. Thus, A is fixed by the numbers (2, 1), B by (16, 5), and C by (7, 15). So, Lf you were told to mark on a 
sheet of st^uared pajjer the points (2, 1). (10, 5). (7. 15), and consider them to be the corners of a triangle, youi- di-awing would be like Fig. 118. 

85. An Irregular Polygon (Hexagon) fixed in a Plane by its Co-ordinates. — You wlU by now see how exceedingly convenient this method of 
fixing the jjosition of points in a plane by means of their co-ordinates is, and you will now be able to solve some problems for yourself. For instance, if 
yon were given a sheet of squared paper, and told to plot an u-regular polygon (in this case a hexagon) whose corners had the following co-ordinates 
(24, 1), (20, 16-5), (13, 13), (10, 16), (3, 10), you would produce a figui-e like that shown in Fig. 119. 



/5 




-20-10 10 40 70 

Iy' 

Fig. 117. —Graph of x = 3y - 20. 



100 



'^^^^ 


Hi 




^Hf = = y 



Fig. 118. — Position of a triangle. 



IS 


1 


















^ 


















3 






























> 


^ 














> 
































/ 






s 


















\ 










13 


























*> 
















1 








































































































L 








10 








/ 










































































































^^ 






















































"«. 


L, 






















































^ 
























































'■ 


> 


















' 






































^ 


















































































































■s. 















































































































?'^ 


X 






3 












(0 




(3 












ZO 






25 



Fig. 119. — Position of an irregular 
polygon. 



86. Plotting Exptrimental Results on Squared Paper. — If you have not already done some work in a mechanics or physics laboratory, you 
probably soon will, then you will find that the results of experiments in meclianical or physical science are invariably, when iiracticable, plotted on squared 
paper, giving what are called experimental curves, and showing at a glance the relations between simultaneously varying quantities which are mutually 
dependent. And you will soon realize that no symbols or figures can possibly convey to your mind so clear an idea of these relations as a simple figure 
plotted from the experimental data. As an illustration of the method, we will proceed to draw a curve to show how the mean velocity of water passing 
through a 10-feet circular se'wer or culvert varied in a particular case with the depth of the water in it. It was experimentally found that, for the 
fractional parts of the fuU depth represented in Columns A, the relative velocities were represented by the quantities in Columns B. 



' In the ordnance and other extensive topographical surveys the area of which a map is to be drawn is divided into squares, which are suiveyed by different 
persons. All the surveys (squares) being afterwards collected and placed together to form the whole area. 



40 



INDUSTRIAL DRAWING AND GEOMETRY 



Table A. 


Table B. 


Table A. 


Table B. 


Table A. 


Table B. 


Table A. 


Table B. 


Table A. 


Table B. 


Table A. 


Table B 


Proportion of 


Relative 


Proportion of 


Relative 


Proportion of 


Relative 


Proportion of 


Relative 


Proportion of 


Relative 


Proportion of 


Relative 


depth of flow. 


velocity. 


depth of flow. 


velocity. 


depth of flow. 


velocity. 


depth of flow. 


velocity. 


depth of flow. 


velocity. 


depth of flow. 


velocity. 


0-00 


0-000 


0-20 


0-696 


0-40 


0-929 


0-60 


1-053 


0-80 


1-100 


1-00 


1-000 


005 


0-367 


0-25 


0-765 


0-45 


0-968 


0-65 


1-069 


0-85 


1-097 






0-10 


0-506 


0-30 


0-828 


0-50 


1-000 


0-70 


1-085 


0-90 


1-088 






0-15 


0-613 


0-35 


0-884 


0-55 


1-028 


0-75 


1-094 


0-95 


1-066 







To set out the curve, take a sheet of squared paper, and from A (Fig. 120), plot the quantities representing- the proportions of depth of flo-w^ along 
the axis OX (marked AB), as shown in the figure. Then along the axis OT (marked AC), at right angles to OX, mark off quantities to represent 
the relative velocities, letting them increase by O'l. 



1-100 

1-000 
■§• 9000 
'^- 8000 
§■7000 
^-6000 
•5- 5000 
■§■4000 
3000 
2000 
■1000 



^ 

































G 






























F 






— 




"~ 


1 


100 


■~' 


^^ 


H 




















^ 


" 


I0< 


















■00 
















E. 


X 






































/ 


y 








































/ 








































D 


/^o. 








































/ 



























































































































































































































•05-I0^l6^20^25^30^35^40^45-50^55^60^65^70^75^80^85^90^95 10 
Proportion of depth of flow 

Pig. 120. — Plotting experimental results. 

The height of a point to represent any actual relative velocity from the table can then be easily estimated and marked on its pro2]er line ; for 
instance, the highest point will be G = 1-100 for a depth of 0-80, so that the intersection of the line throug-h 0-80 and 1-100 wiU g-ive this point, and the 
other points having been similarly found, a fair flowing line through the points wiU be the required curve.' 

In some cases only a few points, such as D, E, P, G, H, need be fixed to get the run of the curve. This graphic method of studying mechanical 
and physical matters has a great additional advantage over any other system, as an error in the determination of any of the quantities is 
immediately detected owing to its point coming- noticeably outside the curve. 



' In examining the curve it will be noticed that it attains its greatest value at 0-80 of the full depth, this value being considerably greater than at full depth. 
This clearly shows that when the sewer is running full, it is not discharging to its full capacity. This is mainly due to the increase in the value of the wetted 
perimeter being much more rapid towards the top than that of the area. 



USE OF SQUARED OR SECTIONAL PAPER 



41 



EXEECISES. 

Suggestions fob Oral Questions. 

1. If in doubt about the accul'aoy of the ruling of your squared paper, how would you test it ? 

2. In fixing the position of a point by its co-ordinates, what is the name of the lines you measure from '? 

3. What name is given to the point where the axes or lines of reference meet or intersect ? 

i. State the form the general equation of a straight line which passes through the origin takes ? 
5. What form does the graph of the equation y = kx take ? 



r 














































































5 — 





























































































































































































































Y|||||i|||||i,f.| 




-.^ "i 
















-tiQ 











mrYii II III 1 Miri 1 


~I 


T 


t 




- - - 2" 


A 


z 


,/• 




B -" - -- - 


K^ a 






"~0 



II III Yliii III 1 |I||B 


-1- -^ 


^' 


^ 


^' 


._ 


-=" 


^' 


h ^"^ 


^' _ _ 










"U " 



Fig. 121. 



Pig. 122. 



Fig. 123. 



Fig. 124. 



6. The large squares on a sheet of squared paper are 1" and the small ones j^" : how many of the small ones are there enclosed in each large one ? 

7. What are the co-ordinates of the point P in the given figure ? (Pig. 121, which may be drawn on the blackboard). What number represents x, and what y? 

Drawing Exercises, etc. 

8. Write down the co-ordinates of the points A, B, C, and D in Pig. 122. 

9. Plot {i.e. mark on squared paper) the following points: (6, 13), (2, 15), (12, 9), (4, 17), (10, 18). 

10. Plot the points (0, 0), (5, 3), (7, 5), (6, 10), (0, 14), (-5, 13), (-11, 11), (-15, 5), and draw a fair line or curve through them. 

11. Write down the co-ordinates of a point P which is 2" from (15, 0) and (0, 5). 

12. The centre of a circle 2" diameter has co-ordinates (8, 5). Find the co-ordinates of the points on the axes cut by the circle. 

13. Plot the line whose equation is y = O'Sa". 

14. Graph the equation y = 0'45a;, and write down the co-ordinates of the point in the line which is 1" from the K-axis. 

15. Graph the line whose equation is a; = 2y — 15. 

16. Draw the triangle whose corners A, B, and G have the foDowing co-ordinates : A (3, 1), B (22,6), G (8, 12). And find the cc-ordinate of a point inside the 
triangle equidistant from its three sides. 

17. Write down a few co-ordinates of the curve ABC, Fig. 123. 

18. A line passes through the origin and is inclined 30° to the x-axis ; find, if you can, the equation of the line. 

19. Find, if you can, the equation of the line AB, Pig. 124. 



CHAPTER IX 



AREAS AND THEIR MEASUREMENTS 

(The definitions relating to areas are given at the end of the hooh) 

87. Introduction. — By this time you are probably aware that the bouudary-line or perimeter of any closed figure encloses an amount of 
surface called its area. The unit of area for ordinary measuring purposes may be a square foot, a square inch, or a square centimetre. 
A square foot is the area of a square drawn on a side one foot in length ; a square inch (Fig. 125) is the area of a square drawn on a 

side one inch in length ; and a square centimetre (Fig. 125) is 
a square drawn on a side one centimetre in length. 

If we had a rectangular floor laid with square tiles, each one 
having sides of one foot, you would speak of the area of each tile 
as being one square foot. Now supposing that on counting the 
number of tiles in one row you find there are 16, and that there 
are 10 rows, you would intuitively know, even if you had not 
been told, that the number of tiles on the whole floor can be 
found by multiplying the 16 by the 10, giving 160. That is 
to say, the area of the floor is 160 square feet.^ Or, in other 
words, the area of the rectangular floor equals L X B, that is, 
its length multiplied by its breadth. 

Now, supposing you were to cover your sheet of paper with inch squares, draught-board fashion, you would have before you a 
sheet of squared paper, as it is called, and you could mark off along one line of them 16, and along a line at right angles to it, 10. 
The rectangle so formed would represent the floor, as in Fig. 126, of course to a scale of one inch to the foot, or to one square inch 
to one square foot. The latter, of course, gives the ratio of the areas, but this gives you no direct idea of how many times larger the 
floor is than your piece of paper representing it. However, if you cut out a piece of tracing paper one foot square, and place it 
over your inch squares and count the number enclosed, you find that there are 12 x 12 = 144, as in Fig. 127 ; that is to say, 

' You have doubtless in buying stamps often counted them in this way : you do not trouble to count each one, you count the number in a row and multiply 
it by the number of rows. Thus, if you were buying five shillings' worth of penny stamps, you would receive 60, and if there were 10 in a row, you would know at 
once that the number was right if you had 6 rows, as 6 x 10 = 60. 

42 




f 1 1 1 1 1 1 1 1 1 1 1 '1 I ' 


^ Area of Rectangle LXB 


1 


^ — — ._ — — ^ 


•- 


(W 




I 


}IL 



Fig. 125. — Comparison of square inch 
and square centimetre. 



Pig. 126.- 



-Area of rectangular 
floor. 



AREAS AND THEIR MEASUREMENTS 



43 



each tile has an area 144 times the area of one of your inch squares. And of course it follows that the scale of areas is 1 square 
inch to 144 square inches, or 1 to 144 ; that is to say, 144 pieces of paper containing 16 X 10 inch squares would cover the floor. 



















































































At 


"€3=/ 


Sa 


Foot 


1 1 


1 


1 ' 


J_L 






























yp-J 


An 


ea- 


k/s 


Q- 


fn 


ch- 


■ 1 1 1 1 r 


LiAjj 




A B 

Fig. 128. — Equal parallelograms on the same 

DEC 



Pig. 127. — Square foot and 
square inch compared. 




G 


!; 


H 


C 

E 


/ 


/T 




A F 


B D 



Pig. 130.- 



-Reotangle equal in area to 
triangle. 



It -n-ill now be helpful to examine a few typical problems relating to areas of simple figures, and you should experience no difficulty in under- 
standing the following, and the bearing that one may have upon the others. 

88. To construct a Rectangle equal in Area to any given Parallelogram. — The working of this problem depends upon the geometi-ical fact 
that parallelograms upon the same or equal bases, and between the same parallels, are equal in area (Euo. I. 35, 36). Thus, the parallelogram 
ABCH (Fig. 128) is on the base AB, and if on this base we construct a rectangle ABEP. the side EP being on the side CH produced, we shall have the 
rectangle (which is also a parallelogram) equal in area to the given parallelogram, as required, because the before-mentioned conditions are complied with. 

Notes. — 1 It will be seen that the parallelogram ABDG is also on the same base and between the same parallels, and is therefore equal to ABCH and ABEP- 
The area of a rectangle, as you are awaro, is expressed by multiplying two adjacent sides together. Thus, area of ABEP = AB x BE. It follows that the area of 
any parallelogram is equal to one of its sides multiplied by the perpendicular distance between that side and the opposite one ; thus, the area of ABCH = AB X BE. 

2. Experiment. Make a large drawing of the figure, and with your scissors cut out the triangle APG. Yon will find that it exactly covers the triangle BED. 
Treating BEG in the same way, it will cover APH. You will know what importance to attach to this experimental proof, which with a little thought can be 
applied to other oases, such as the next problem, for instance. 

89. To draw an Isosceles Triangle whose Area shall be half that of a given Rectangle.— Let ABCD (Fig. 129) be the given rectangle. 
Bisect AB by the perxiendicular FE, cutting CD in E. Join AE and BE ; and AEB is the required triangle. 

It ■n'ill be at once obvious that the rectangle ABCD is cut into two equal parts by EF, and also that the diagonal AE cuts the rectangle AFED into 
two equal parts ; and BE cuts FBCE into two equal parts. Therefore the area of the triangle AEB (made up of AEF and BEF) is equal to the sum 
of the areas AED and BEC ; that is, it is equal to half the rectangle. And as AE and BE are equal, AEB is an isosceles triangle. 

Note. — The diagonal AG cuts the rectangle iuto two equal parts, and so does BD. Therefore the triangles ADB and ABC are each equal to half the rectangle, 
and therefore must equal the triangle AEB. It will be noticed that the three equal triangles are on the same base and between the same parallels. Euclid (I. 37) 
proves that all triangles on the same base and between the same parallels are equal in area. 



44 



INDUSTRIAL DRAWING AND GEOMETRY 



90. To construct a Rectangle equal in Area to a given Triangle. — If you understand the previous problem, the working of this one should be 
obvious. 

Let ABC (Fig. 130) be the given triangle. At A and B erect AT and BJ. perpendiculars to AB. From C draw CD perpendicular to AB (or AB 
produced) and cutting it in D ; bisect CD (the altitude) in E, and draw BJI parallel to AB, and cutting AI and BJ in I and J. Then ABJI is the 
reo[uii'ed rectangle. (Based on Euc. I. 41.) 

Note. — Prom this it will be seen that the area of a triangle may be expressed thus : Area = base x i altitude ; or, obviously, the altitude multiplied by half 
base will also give the area. The dotted rectangle BPGH illustrates this. 

91. To construct a Rectangle equal in Area to a given Rectangle, One Side of the former being given. — The solution is not so obvious as 
that of the preceding problems, and you need not att«mpt it if you are reading the chapter for the fii-st time. 

Let ABCD (Fig. 131) be the given rectangle, and AE the given side. Then the other side of the required rectangle is a foirrth proportional to AE, 
AB, and AD. Set off along AB from A the given side AE ; join E and D, and through B draw BF, cutting AD produced in F. Tlien AF is the 
fourth proportional and the required side. Complete the rectangle by drawing FG and EG parallel to AB and AF respectively. 

Pkoop. — By similar triangles, AB ; AP : ; AE ; AD, .•. AB x AD = AP x AE ; that is, the two rectangles are equal. 

Note. — This problem is of great importance when moments have to be manipulated for the geometrical determination of the centre of gravity of figures. It is 
also a simple way of finding a line to represent a rectangle, i.e. if AE = 1", then AP would equal the area of ABCD in square inches. 

92. The Pythagorean Theorem. — If you draw a right-angled triangle (Fig. 132) making its sides in the ratio of 3 : 4 : 5, as 
shown, and construct on each side a sqiiare, and divide each of the squares into smaller squares whose sides are the equal divisions 

of the lines, the square ACGH, you will find on 
counting, contains nine small squares, and ABFE 
sixteen squares, whilst CBJK, the largest one, con- 
tains 25. You will be struck with the fact, of course, 
that 25 = 9 -(- 16, that is to say, the area of the 
square on the hypotenuse of the right-angled triangle 
equals the sum of the squares on the other two sides. 
Indeed, this theorem, which was discovered by Pytha 
goras about 580 years B.C., and applies to all right- 
angle triangles, is proved in the 47th Prob. of Euclid's 
first book. 

The isosceles triangles in Fig. 133 also shows this beautiful relationship between the areas of the squares ; for obviously all 
the triangles into which the squares are divided by the diagonals are similar and equal, therefore the square on CB equals the sum 
of the squares on AB and AC, as there are four triangles in the former and two in each of the latter. You should draw this 
figure to a large scale, and cut out the triangles on the small squares, and place them together to cover the large square. 

93. To construct a Square eqaal in Area to the Sum of Two given Squares. — You will now experience no trouble in working this 

and similar problems. Let the given squares be of If" and 1" sides. Draw EC (Fig. 134) If" long, and AB 1" long, perpendicular 

to BC. Join AC, and with AC as side construct a square ACDE, which is the one required. 

Note. — This problem is of great practical importance, as it equally applies to all similar figures, and you should endeavour to understand that the areas of 
similar figures are to one another as the squares upon their corresponding sides. A glance at the dotted triangles and hexagons on the figure will help you to under- 
stand this truth, which is further illustrated by Prob. 96. 







■ 




VXA 


- 




- 


^ 


H A 








































E 











G C 


/ 1 

/ J 


\ 
\ 
\ 


\ 


H Al 
E 


\ 

\ 
\ 





Pig. 131.- 



-Bectangles of equal 
area. 



Pig. 132. 



Pig. 133. 



AREAS AND THEIR MEASUREMENTS 



45 



94. Area of a Circle. — If you divide a circle (Fig. 135) into, say, 24 equal parts or sectors as shown, and cut these parts out with 
a scissors, you can arrange them as in Fig. 136, and see that they form a figure which in slaape is very nearly a rectangle. It will 
occur to you, of course, that the larger the number of divisions you make, the more the figure will approacLi an exact rectangle. 

The area of the rectangle equals that of the cu'cle, of course, and we know that the former = base x height = ttR x K, or ttR^ 
= the area of the circle. 

This will conveniently lead up to the next problem. 

95. To construct a Rectangle equal in Area to a given Circle.^ — Let the circle (Fig. 137) be the given one. As you have seen, 

a rectangle whose short sides are equal to the circle's radius, and whose long sides are equal to 

^,'~"-.^ ttR (or say 3| the radius), will equal in area the circle. Then, such a rectangle may be drawn 

/■^ "~-\^ by taking the radius AB as the short side, and at B setting off BC at right angles, and equal in 

length to 3f AB. Complete the rectangle by making AD and DC parallel to BC and AB 

respectively. 

Notes. — 1. A triangle equal in area to the circle may, of course, have a base equal to BC, and an altitude twice 
AB (equal to the circle's diameter), or it may have AB as altitude and base twice BC. 

2. A second method of determining a line equal in length to half the circumference is shown by the dotted lines 
on the figure. AP is drawn making 30° with AE, and cutting a line through E parallel to BC in P. Marli a point G 
three times the radius of the circle from B, and the distance EG is a close approximation to ttR. 

96. To describe a Circle equal in Area to the Sum of Two given Circles. — Let the given circles 




Fig. 134. — Figures equal in area to sum 
of two similar figures. 




■Figure equal in area 
to circle. 




-Rectangle equal in area to 
circle. 



be of 1" and If" diameters. Draw BC (Fig. 138) If long, and AB 1" long, perpendicular to BC. Join AC, and with AC as a 
diameter, describe a circle, which is the one required. 

Notes. — 1. The areas of circles vary in the same proportion as the squares on their diameters. Thus this problem is similar in principle to Prob. 93. 

' A simple practical test for the equality of the areas (if you have been using a sensitive balance) is to carefully draw the figures (the circle and rectangle) upon 
a piece of millboard or thin sheet metal, and to cut them out and weigh them. The equality of the area of the circle to that of the rectangle ABCD can then be 
seen. If you draw the figures on squared paper you will not fail to notice that the rectangle is equal to 3| squares, whose sides are R, and that therefore the 
area of the circle = 3iR-. Of course 3| is only an approximate value of ir, as we have seen. 



46 



INDUSTRIAL DRAWING AND GEOMETRY 



2. The areas of all similar figures vary in a like way, as we have seen ; thus, if pentagons were constructed on AB, BC, and AC, the one on AC would be 
equal in area to the sum of the other two. 

3. This problem is of practical use when it is required to find the diameter of a pipe whose sectional area is equal to that of two others, i.e. one which 
will contain the same quantity of liquid as the two others together. An obvious variation of the above construction enables you to find a circle equal in area to 
the diSerence of two others. 

4. A very elegant variation of this problem is Hippocrates' theorem of the lunulse (or lunes). The semicircles being similar figures, by the I. 47, the 
area of the semicircle ACB (Fig. 139) is equal to the sum of the areas of the semicircles ADC and CFB ; if from the equals we take away the crosshatched 
parts AEC and CGB, the remainders will be equal. That is, the area of the triangle ACB will equal the sum of the areas of the lunes ADCB and CPBG. 

5. It follows from the preceding note, that if the triangle ACB (Pig. 139) be made isosceles, i.e. the angles BAG and ABC 45°, each of the lunes will have an 
area equal to that of the square on the radius of semicircles ADC or CPB. Draw such a figure, and satisfy yourself that it is so. In Pig. 140 we have half such a 
figure, CE being the axis of symmetry. Here we have the direct construction for making a lune APEGA equal in area to that of a given square BCDA, where 
the radius of the semicircle is AD, the side of the square ; and the radius of the arc AGE is CA, the diagonal of the square. 

97. To draw a Square whose Area shall equal, say, Five Square Inches. — The direct method of working this problem is to 
construct a square upon a line whose length is equal to the square root of the given area. The length of this line will equal \/5 . 
Then draw a line, AB (Fig. 141), 2" long, and at A erect the perpendicular AC, 1" long. Join BC, and the length of this line will 
equal ^ \/5. Upon this construct the required square. 





Pig. 138. — Circle equal in 
area to sum of two others. 



A B 

Pig. 189.— Area of the triangle equals 
sum of areas of the lunes. 





Fig. 140. — Lune equal in area PiQ. 141. 

to the square. 

■Let ABCD (Fig. 142) be the given rectangle. Produce BA to E, making AE 



98. To make a Square equal in Area to a given Rectangle.- 
equal to the side AD. 

Find a mean proportional (AF) of AB and AE (Prob. 64. Also refer to Euc. II. 14, III. 35). Then upon AF construct the required square. 

Note. — If the side AD of the rectangle be made equal to 1", then, as we have seen, the length AB of the other side in inches will represent the area of the 
rectangle, also that of the square, and AF will equal the square root of theTiumber representing the area. 

' You will remember that the length of the side of any square is equal to the square root of the area of the square. For example, a square whose area is 
4 square inches will have sides whose length is ^4" = 2", and one whose area is 1 square inch will have sides = Vl" = 1", and, again, one whose area is f square 
inch will have sides = V J" = i". Bearing in mind these facts, and making use of Prob. 97, you will be able to draw a square of any given area, or any line whose 
length is a root quantity, expressed thus Vn, where N is known. 



AREAS AND THEIR MEASUREMENTS 



47 



99. To reduce any Irregular Figure to a Triangle of equal Area — Let ABODE (Fig. 143) be the figiu-e. Join AD, and througli B tha.w a 
Hue i^arallel to AD. cutting AB in F ; join DF. Similarly, join BD, and through C draw CGr imrallel to DB, cutting AB produced in G. Join 
DG. and the triangle FDG is equal in area to the given figure. 

This will he readily understood when it is seen that the triangles BDC and BDG are upon the same base, BD, and between the same parallels, BD 
and CG, and are therefore equal in area (Prob. 89, and Euc. I. 37) ; that is, we have cut off the triangle BDC. and put on one, BDG, equal in area, and 
therefore have not altered the area of the figure ; and, in the same way, at the other side of the figure, we have replaced the triangle FEA, on the base 
FE. by the triangle FED, on the same base and between the same parallels. 

Note. — This principle once thoroughly understood, the most complicated rectiliueir figure can be easily reduced to a triangle. 



100. Measurement of Land, 



^ 


D ^\ 


r 


\ 



A 

Fig. 



B 



142. 




Land Surveying 

Surveyors measure land with a chain invented by a Mr. Gunter, and therefore known by the name 
of Guntcr's Chain. It is 22 yards, or 4 poles, in length.^ and is divided into 100 equal parts, 
called links, each link being 7' 9 2 inches. 

100a. An Acre of land is equal to 10 square chains ; that is, a strip of land 1 chain in 
breadth and 10 in length (Fig. 144) has an area of 1 acre ; and this equals 22 x 220 = 4840 
square yards (= 66 X 660 = 43560 sq. ft.); or 4 x 40 = 160 square rods, or perches; or 
100 X 1000 = 100,000 square links. 

It is usual to give the measurement in acres, roods, and perches : 4 roods being an acre, and 
40 perches a rood. 

A statute-pole is 16.V feet long : but in different parts of the country there are by custom 
poles of different lengths — 21, 18, 15 feet, etc. 

101. The Land Surveyor's Operations. — The fundamental principles underlying the operations 
in any survey are the same. There are three separate operations :— 




F B G 

Fig. 143. — Triangle equal in area to 
irreffular figure. 



70 CJvojtns =70x 66^ 660 ft. 

Pig. 144. 



-^ 



1st. The taking of the measurements ^ on the ground. 

' The chain used by civil engineers is 100 feet in length, with 1 foot lints. 

- The origin of the science of geometry appears to be due to the efforts made in the early ages to measure laud. This naturally suggested problems on the areas 
of triangles and other simple figures ; indeed, the earliest solutions of such problems appear to be afforded by a Papyrus in the British Museum, giving rules for 
the calculation of triangles, trapezoids, and circles, which is believed to have been copied from a much older work about 1700 years before the Christian era. 



48 



INDUSTRIAL DRAWING AND GEOMETRY 



2nd. Making the drawing or plan. In other words, plotting the measurements on paper. 

3rd. From such plans or drawings measuring the areas, or arranging the work for which the survey was made. 

Measurements in the field are taken either by linear or angle measuring instruments, or by a combination of laoth ; l3ut whatever 
the system of measurement may be it is a necessary condition of good practice to check all measurements (with the exception of 
offsets of a few links in length) by other linear or angular measurements. 

Having made these passing remarks on a branch of work that has much to do with the geometrical side of our subject, we may 
pass on to the examination of a typical example or two. 

102. The Field-book.— The usual method of entering the field-notes is to legin at the bottom of the page and write upwards. Each 





Field-book 




i Links. 


go East 


C 722 
From 


To B 
1 1456 

882 
A 

1 




V- 



- £82 — -^ 
- — — - /4S6 — — 
Fig. 145.— Field of three sides. 



page of the field-book is divided into three columns. In the middle column is set down the distances on the chain line, at 
which any offsets, marks, or observations are made, and these offsets, etc., are entered in the right and left-hand column, as you 
will see. 

103. To measure a Field of Three Sides. — Let ABC (Fig. 145) be the field. At each corner of the field place a station 
staEf.i Then, in chaining from A to B (going east), measure till you find c, from which a perpendicular will rise ^ to the opposite 
corner C, and enter in the field-book the distance from the end A to this point, namely, 882 links, and then continue the 
measurement of the line, 1456 links ; this length being the distance from A to B. The perpendicular (or ofifset) is now measured 
and found to be 722. 

Centuries later the annual overflow of the Nile, and the consequent destruction of landmarks of the different proprietors who paid tribute to the king, led to 
the necessity of measurements being made each year to restore the marks, and gave an impetus to the study of simple geometrical problems. 

The only geometry known to the Egyptian priests was that of surfaces, together with a sketch of that of solids, a geometry consisting of some simple 
quadratures and cubatures which they had apparently obtained empirically. 

The ancient Greeks too, judging from the problems discussed by Hero of Alexandria, were led to study the science in connection with the surveys of 
their mines. 

' A pole some 6 ft. in length, usually painted in red, black, and white, in broad alternate bands. Used with a red or white flag, when the poles are placed far apart. 

^ The exact position of c along the line AB is generally found by using a small instrument called an optical square. 



AREAS AND THEIR MEASUREMENTS 



49 



To Calculate its Area proceed as follows : — 



Calculation. 


Triangle ABC. 


1456 Base AB 
772 Ofiset Cc 


2912 
10192 
10192 


11,24032 Double area 


5,62016 

4 


2-48064 
40 



9-22560 Ans. Area = 5 a. 2 r. 9J p. 



104. To measure a Field with several Sides. — If the field is 
somewhat elongated, as in Fig. 146, it is often convenient to 
choose the longest diagonal AE, as a base line. Then, after 
fixing station staffs or poles at the corners A, B, C, D, E, F, and 
G, to chain the line AE, measuring till you find a position from 
which a perpendicular will rise to the first corner B, and enter 
the measurement, as explained in Art. 102, and shown in the 
figure (146) wliich accompanies the field-book. Complete the 
survey by repetitions of this operation for the corners G, C, F, 
and D. 

105. Offsets, and how to measure them. — It will be noticed 
that in Figs. 145 and 146 the straight lines which are 
measured in the field do not exactly coincide with the uneven 
boundary line of the field. This necessitates occasional offsets 
being taken, to the right or left as may be required, and these 
offsets are measured by what is called an offset-staff, a round 
wooden rod, usually 10 links in length. The offsets are entered 
in the field-boolv by taking a separate column for each side, 
AB, BC, etc., of the field, and entering the oifsets as we did 
in connection with Fig. 146. 

r 



PiBLD-BOOK. 





Links. 






ToE 






1263 






1096 


D142 


P156 


989 






863 


C254 


G280 


581 






456 


B120 


Prom 


■ A 


go North 




50 



INDUSTRIAL DRAWING AND GEOMETRY 



106. To find the Area of a Rectilinear Enclosure in Sc[uare Feet. — Let Fig. 147 be the enclosure, with the measurements in feet/ 
tabulated in the field-book as in Ai-t. 105. Then we have : — 



Field-book. 





Feet. 






ToD 






140 






120 


C35 


E40 


90 






30 


B50 


From 


A 





Calculation op Abea. 

A A6B = iA6 X 6B = i X 30 X 50 = 750 

Trapezium 6B0c = fee X i(6B + cC) = 90 x i(50 + 35) = 8825 

A cCb = icD X cC = i x 20 X 35 = 350 

A ADE = JAD X cE = J X 140 X 40 = 2800 




Area = 7725 sq. feet 



Fig. 147. — Rectilinear enolosm-e- 
That is, the required area is 7725 square feet. 
107. To draw a Square or any Regular Figure equal in Area to a Closed Figure bounded by an Irregular Curved Line. — The 

irregular figure shown in Fig. 148 may be supposed to be the plan of a field, which has been drawn 

from notes made in the field-book from the surveying operations. The object of the problem is to 

find a square or rectangular plot equal to it in area. ' You may take a piece of tracing paper and diuw 

across it a fine straight line, and place it over the figure (Fig. 148) and prick off the positions of such 

lines as AB, BC, CD, and DA, being careful to place them so that the lines give-and-take ; that is, cut 

off as much of the figure as they add to it. Most students have an eye true enough to enable them, 

with care, to draw in this way a straight-line figure (or polygon) closely approximating to the curved 

one. This figure can be divided into triangles, and these in their turn converted into rectangles 

and equivalent squares, etc., by the previous problems, as may be required. 

Note. — Tlie area of the figure can be accurately measured by the use of an instrument called a planimeter, and its 
perimeter can be approximately measured by rolling along the boundary a coin or the lid of a piU-box, taking account of the 
diameter of the rolling circle and the number of revolutions. 

Pig. 148.— Irregular figure. 108. To find the Area of a Figure bounded by Three Rectangular Lines and an Irregular Curved 

Line.'^ — Tikst Method. — Draw a fine line on a piece of tracing paper, and hold the paper over the 
curved line EC (Fig. 149) till the straight line EF becomes a give-and-take line ; that is to say, place it so that it cuts off as much 

' Measured by using an engineer's chain of 100 feet and a tape measure for the offsets. 

''■ You will ere this have noticed that it is only in the ease of a few regular figures that the area is connected by any simple relation with the linear dimensions, 
so that it can be calculated from those dimensions. In a great many difficnlt cases the following practical method of measuring an area can be adopted with 
advantage. Draw the figure on a sheet of cardboard, miUboard, or thin sheet metal, cut out the figure and weigh it in a suitable balance. If you know by 
a separate experiment how many square units (centimetres or inches) weigh one gramme, you can thus find the area of the figure which has been cut out. 




AREAS AND THEIR MEASUREMENTS 



51 



of the figm'e ABCD as it adds to it, as in the previous problem. A little careful judgment will enable you to do this with a degree 

of accuracy near enough for many practical purposes. Having fixed the line, prick off the points E and F where it cuts AD and BC. 

AE + BE 
Then, obviously, AB x ^ will equal the area of the given figure. 

109. Another Method of measuring Areas. — Given a Closed 
Figure bounded by an Irregular Curved Line, representing a Field to a 
given scale (say, 3 chains to the inch), to determine its Approximate Area 
in Aeres.^ — This required area could be determined by Prob. 107, by first 
finding an equivalent figure bounded by straight lines, and then by dividing 
the figure as found into triangles, or triangles and trapezia, whose areas 
could be readily measured by using a chain scale, and by adding the 
areas of the triangles, etc., to determine the whole area. But a very 
simple, practical way of determining the area is to set out a number of 
parallel lines on a sheet of tracing-paper, making the distance between 

10 10 10 

them = 



(No. of chains to the inch)^ 
chains to the inch, to which the figure is drawn, 
tracing-paper is placed over the figure (as in 
Fig. 150), the lengths of equivalent strips of 
width ly into which the figure is divided 
can be measured (as shown) by a rule (or, 
better still, a strip of paper, ticking off the 

^ — c 



^ = -p- = ly for the scale of three 
Now, if this sheet of 






Fig. 149. — Example of a give-and-take line. 



Fig. 150. — Measure of acreage. 



Fig. 151.— Area of an indicator diagram. 



separate lengths), and each inch in length will be equal in area to one acre. 

110. Given a Closed Figure bounded by an Irregular Curved Line, representing an Indicator Diagram from a Steam Engine, to 
determine its Approximate Area in Square Inches. — As will be seen from Fig. 151, the expedient employed in the previous problem 
may be used, making the distance between the parallel lines on the tracing paper (or celluloid), ^' , then the sum of the lengths 

' This ready way of determining such an area is sometimes practised by surveyors, when the figure represents a plot of land. 



S2 



INDUSTRIAL DRAWING AND GEOMETRY 



of the parallel strips divided by four will obviously give the area of the figure in square inches. As a matter of fact, the sum 
of the lengths in this case is 15'375", as you will find, if you carefully measure the lengths of the strips. So that the area = 15"375 
-1- 4 = 3'844 square inches. 

Note, — The error due to want of accuracy (the personal equation) should not exceed 5 per cent. 

111. Area of Figures on Squared Paper. — The following typical cases should help you to measure areas of figures on squared 
paper. The figures may be either drawn on the squared paper, or squared tracing paper may be used to place over the figure or over 
part of a drawing. 

{a) When a Side or Diagonal of the Figure can \e made to coincide with a Line of the Squared Paper. — The area of any rectilinear 
figure ABODE (Fig. 152) can be found with the aid of right-angled triangles and rectangles. The given figure, you will see, is 



^I _1_ 




A I8"""~'~- 


/ 1 \ 


t^ 4V 


^ -u Ha 


^ ' Oft. _|J 


1 \ 96 j \ 




/22 1 -gt 


i^± :--J-H>^ 


^ ^---^ 27>^ -T- 




■?B 



f'|- ■ 


--du 


^. 




- — 








- jf 


^ 












Zlj - 


^" 






- — 


i 




Tat - 


■ *■ 


\ 




s9 '^ 




\ 






1 




t ' 






^. 




1 






95^ 




s., 






" 1 /I 1 1 1 






s 


1 


1 f 










\ 


1 


1 / 












Vifi 


' 








__!- 




r 


J 




"___ - 








Af=- 


'-rZ. .. 


-- 


-JD 





.-'] sy\~^~' 


INI 


• 


'■ \ 9]] ^Nl 1 1 1 


/ 1 M I/2II 1 M 


\ 


I 


^3 _ //T> 


^ 


J 


,4. -102 


^ 


I 


16 


!^ 




16 




D 


. ~M 


F 








aV- 


7^ --; 


■-/c 


s ^ 


tSM- 


z 


\ 


/2W54 


z 


^-J 1 1 1 Sll , 1 Uf 1 1 1 




l-^l eJU^I 


ITI 1 



Fig. 152. — Area on squared paper. 



Fig. 153. — Area of triangle. 



B 

Fig. 154. — Approximate area of segments. 



divided into four triangles and a rectangle, the numbers marked on the figures indicating the number of squares each contains. 

Thus the area of the whole figure equals 22 + 27 + 8 + 18 + 96 = 171 squares. Now, if the sides of the squares be 10 to the inch, 

171 
obviously the area of the figure equals — rg = 1-71 square inches. 

(J) When an Important Line of the Figure does not coincide with a Line of the Squared Paper. — As in Fig. 153, it is convenient to 
draw rectangular lines outside the figure, making, if possible, a rectangle (ADEF) whose area can be readily determined (in this case 
17 X 13 = 221), then by subtracting the right-angled triangles ADB, BEG, and CFA from the rectangle, the required area of ABC 
is found ; that is, area of ABC = 221 - (39 -f 55 -|- 31^) = 95| squares. 

(e) When the Figure is Curvilinear. — Although the areas of such figures cannot be found exactly by the method of counting 
squares, approximate values may be easily determined in the following way. 

Let the figure be the segment of a circle ABCA (Fig. 154). In counting the squares covered by its area you will notice that the 



AREAS AND THEIR MEASUREMENTS 



53 



arc ABC cuts through several squares, and the question arises, how are you to deal with these broken squares ? Of course you could 
estimate the fractional value of each one included in the boundary line or arc, but this would be a tedious proceeding ; however, 
a fairly correct value can be found by counting 1 if the hroJcen square is more than, half a complete square, and counting if less than 
half a square. Applying this rule to the figure, we find that its area is 54 squares. 

Now, of course, if the segment had been a semicircle, we could more easily calculate its area, arithmetically, by first finding the 
area of the circle, whose diameter in this case would be represented by 16, the area being E^ X tt = 8^ X ir = 64 X 3'14159 
= 201'062, and half this, or 100-531 squares, is the ai-ea of the semicircle. But applying the method of counting squares as a test, 
as in the upper half, DFED, of the circle, we find that the area is 102, as shown, or roughly 1^ per cent, over the true value. 



EXERCISES. 

Typical Obal Qubstions. 

1. What is the name given to the length o£ the boundary line of a plane figure ? And what name is given to the extent of its surface ? 

2. What do you understand by the term area of a plane figure ? 

3. Does a figure with a large perimeter necessarily have a large area ? 

4. What are the units by which areas are usually measured ? 

5. The area of a triangle is i square inches, and its altitude is 3". What will the length of its base be ? 

6. What's the shape of the triangle which has the largest area for a given perimeter ? 

T. A square has an area equal to the sum of the areas of two other squares, one on a 3" side, and the other on a i" side. What would be the area of the square, 
and the length of its side ? 

8. Enunciate the Pythagorean theorem. 

9. The sides of a right-angled triangle are 2" and 3". Give the length of its hypotenuse as a root quantity. 

10. The adjacent sides of a rectangle are 2" and 8". What is the length of its perimeter ? What is the length of the side of a square equal to it in area ? 

11. What is the unit of area in measuring land ? 

12. Give the lengths of the sides of a rectangular piece of land whose area is one acre. 

13. What is the length of Gunter's chain used for measuring land ? 

14. A strip of land 2 chains in breadth is 1 acre in area. What is its length ? 

15. Your cricket pitch is 22 yards long, and your average step 1 yard. You are allowed to use half an acre of land each side of the pitch. How many steps 
away from the pitch would give you the approximate boundary of the acre ? 



Drawing Bxebcises. 

Note. — In working the following exercises it will sometimes be convenient to prick off corners 
of the figures (or other suitable points in them) by placing a page of this Ijook over your 
drawing paper. 

16. Carefully measure the rectangle (Pig. 155), and give its area in square centimetres. 

17. Draw a rectangle equal in area to the given parallelogram (Pig. 156). 

18. Draw an isosceles triangle equal in area to the given parallelogram (Pig. 156). Note. — You 
are to first draw figures like this full size from the dimensions. 

19. Draw a square equal in area to the sum of the two squares (Pig. 157). 

20. Draw a square equal in area to the cross-hatched figure (Pig. 158). 

21. Draw a rectangle approximately equal to the given figure (Pig. 159) when drawn full size. 

22. The radii of an annulus are 2" and 1". Draw a circle equal in area to the surface between the two circles. 



^ 



Pig. 155. 



Pig. 156. 



54 



INDUSTRIAL DRAWING AND GEOMETRY 



23. Draw a square equal in area to the rectangle (Fig. 155). 

24. Draw a triangle equal in area to the quadrilateral (Fig. 160), making A its apex. 

25. The hase of a triangle is 4" and the other sides make angles of 35° and 45° with it.: determine the area of the figure, and check by calculation. 






Fig. 157. 



Fig. 158. 



FiQ. 159. 



Fig. 160. 



26. Two sides of a triangle are 3" and 4", and the included angle 40° : determine the area of the figure, and check-by calculation. 

27. Three sides of a triangle are 3", 4", and 2J" : determine the area of the figure. 

28. Draw a rectangle equal in area to the given figure (Fig. 161), making AB its base. Note. — Refer to Problems 99 and 90. 

29. Convert the irregular hexagon (Fig. 162) into a rectangular figure. 

30. Carefully make the following measurements of the triangle ABC (Fig. 163) : (a) the angle <;> ; (b) the side CB ; (c) the area in square centimetres. 

31. You are to plot the triangular field from the given note below from a field-book (refer to Art. 102), and measure its area in acres, roods, and perches. 
Scale 1" to the chain. 







Links. 




From 


ToB 

1264 
782 
A 


C456 
go West 



Field-book note. Question 31. 



Pig. 161. 



Fig. 162. 



AREAS AND THEIR MEASUREMENTS 

32. The field notes of a six-sided field are given : plot (or draw) the field, and measure its area. Scale 1" = 1 chain. 



55 






Links. 






ToD 






1070 






828 


C40 


E50 


762 






436 


B108 


P84 


315 




Prom 


A 


go North 

1 



Pig. 163. 



Pield-book note. Question S 



Fig. 164. 



33. The plan of a piece of land is given (Pig. 164), it is drawn to the scale of J" to the chain. Measure as nearly as j'ou can its area in acres, etc. 

34. Carefully set out the given figure (Pig. 165), and measure its diagonal BD, and the area of the figure in square inches. 

35. The plan of a piece of land is shown (Pig. 166). AD is a base line, and the angles at the base have been measured, also the lengths of two side?. BG is 
measured on the ground as a check. What should its length be, if all the measurements are correct '? The scale that was used in plotting the figure is 1" = 1 chain. 





Pig. 165. 



Pig. 166. 




36. Carefully draw the figure (Pig. 167) from the dimensions given, and measure and write down the length of the diagonal AC. .\lso measure with your 
protractor the angle at C, and check it by calculation. 

Note. — The sum of all the interior angles of a polygon is equal to twice as many right angles, less four, as the polygon has sides. 

37. Set out the figui-e (Pig. 168) with great care, and measure the diagonal AC, the angle at C, and the distance of P from B. 

38. Measure the area of the sector (Fig. 169) in square centimetres, after carefully reproducing it on your drawing paper. 

39. Make a drawing of the section of a buttress wall (Fig. 170), scale half inch to the foot, and calculate the weight of the wall per foot run, assuming the 
weight of a cubic foot to be 140 lbs. Note. — You will be able to measure the area of the section in square feet. The product of this area and 140 gives 
the weight. 



56 



INDUSTRIAL DRAWING AND GEOMETRY 



40. The section of a caet-iron bearer bar is given (Fig. lYl). Draw it full size, and calculate the weight of the bar : its length is 36", and the weight per cubic 
inch may be taken to be J lb. Note. — AB is the bottom of the section. 



h-'s^ 





Pig. 168. 





Fig. 169. 



•^ — 


'y->\^ s'-^^ 


% 



Fig. 170.— Buttress wall. 



Fig. 171. — Section of cast-iron bearer. 



Fig. 172.— Plan of a floor. 



41. The figured plan of a floor is given in Fig. 172. Make a drawing of it to a scale of ^" to 1 foot, and calculate the number of square yards of linoleum 
that would be required to cover the floor. 



CHAPTER X 

REDUCING AND ENLARGING FIGURES, ETC. 



112. Introduction. — As the practical draughtsman is sometimes called upon to reduce or enlarge figures, we may with 
advantage give a little attention to the expedients usually employed in such operations. If you happen to have a good 
pair of proportional compasses, you can by means of this simple instrument reduce or enlarge drawings so that all the 
lines of the copy shall bear any required proportion to the lines of the original drawing. We will briefly describe 
the instrument, and work just one problem to show how it is used. 

113. Proportional Compasses. — The ordinary form of this instrument is shown in Fig. 173. It can be either set 
to deal with Iines,i or with areas. To set the instrument you must first accurately close it so that the two legs appear as 
one, the nut C of course being unscrewed ; you next move the slider (attached to the nut) until the line across it 
coincides with any required division upon either of the scales ; you now tighten the nut, and the compasses are ready 
for use. It should be mentioned that proportional compasses can also be used to inscribe regular polygons in circles, 
and extract the square roots and cube roots of numbers, but no one troubles about using this instrument for such 
purposes now. Further, it is only of use for any purpose when in perfect adjustment and in skilful hands. 

114. To reduce or enlarge the Lines of a Figure, using Proportional Compasses. — Let us suppose that the irregular 
polygon ABCDEF (Fig-. 174) is the iigure which is to be reduced to a similar one whose sides shall be, say, half those of the given 
one. From any corner F draw lines to corners B, C, and D, dividing the figure into triangles. Now set the compasses so that the line 
across the slider coincides with the division 2 on the scale of lines. The points A, B of the instrument (Fig. 173) will then open 
to double the distance between the points M, N (Euc. VI. 4). Next open the points A, B to the length of the side FA (Fig. 174), and prick 
off Fa with the points M, N, making Fa half FA. In the same way find the points b, c, d, and e, and join these points to fonn the 
reduced copy, ahcdeF, of the given figure. Obviously, if ahcdeV had been the given figure, and we had to enlarge it by doubling the 
lengths of its sides, we should in drawing the lines through the comer F produce them beyond the corners of the given figure, and apply 
the points M, N of the compasses to the given lines, the distance between the points A, B giving the lengths of the corresponding lines in 
the enlarged figure. 

Of course you will understand that in enlarging figures any error made in measuring a line to be enlarged will be proportionally 
increased in the new figure. We may now work a few cases by ordinary geometrical methods, commencing with the one we have just 
worked by using the proportional compasses. 

115. To reduce a given Irregular Figure to a similar one whose Sides shall be, say, half of those of the given one. — 

' A simple form of proportional compasses is made called wholes and halves, because the longer legs are twice the length of the shorter ones. 
This instrument is also useful for dividing lines by continual bisection. 

57 



Pk 



A B\| 

Fig. 173. 



58 



INDUSTRIAL DRAWING AND GEOMETRY 



First Method. — Let ABCDEF (Fig. 174) be the given figure. First divide the figiu-e into triangles by drawing lines FB, FC, and FD through any 
corner, F. Then from F along FE mark off Fe equal to half FE, and through e draw ed parallel to ED and cutting FD in d, and complete the required 
figure by drawing dc, ch, and ha parallel to DC, CB, and BA, as shown in the figure. 

NoTB. — Of course you will remember that the areas of all similar figures are as the squares on their corresponding sides (Prob. 93) ; therefore the area of the 
reduced figure equals one quarter the area of the given one, as shown by the squares EFGH and efgh in Pig. 175. 

You should also remember that the corresponding sides of similar figures or polygons are proportional. Thus, ah '. AB : : aP : AF, or the ratios of pairs of 
corresponding lines are all equal. 

Second Method. — Let ABCDEF (Fig. 175) be the given figure. Then draw ah equal to half AB anywhere parallel to AB, and join Ka and 




REDUCTION" OF riGtJEES. 



Fee 

Pig. 174. — First method. 

B6, and produce the lines to meet in P. 




Fig. 175. — Second method. 





176. — Variation of 
second method. 



Pig. 177. — Method giving inverted 
figure. ■ 



And 



Through P draw lines to C, D, E, and F. Then through h draw be parallel to BC, and cutting PC in c. 
by repeating this operation complete the similar figure abcdef as shown. 

116. Variations of the Previous Methods. — Obviously, if the point P (Fig. 175) be placed 
inside the figure, as in Fig. 176, the reduced figure ahcdef can be drawn as shown. 

Or, if the lines through P be produced as in Fig. 177, the reduced copy can be drawn the 
other side of P as shown. Of course the new figure then becomes inverted. 

117. To reduce a given Figure bounded by a Curved Line to a similar one of Fixed 
Size. — The principle employed in working a problem like this can be readily luiderstood by 
examining Fig. 178, where the figure ABCDEF is reduced to ahcdeF, by first drawing from P 
lines FD, FC, and FB, to any suitable points D, C, B in the curve, and from a point a, which is 
fixed by aA, the amount of reduction required, draw a parallel to AB, cutting FB in 6, and from 
6 draw he parallel to BC, and cutting FC in c, and so on, to complete the figui-e ahcdeF, the 
curve being afterwards drawn through the points a, h, c, d, and e. 

_ Note. — Of course the curve only might have been given to he reduced. The point P would then 
be any point taken at pleasure, and the same construction could be employed. 

118. Reducing and Enlarging Figures by the Use of Squares. — Complicated figures can be readily reduced or enlarged by first drawing over 




REDUCING AND ENLARGING FIGURES, ETC. 



59 



the figTire a number of squares, as shown in Fig'. 179, or placing' over it a sheet of squared tracing paper. It is then only necessary, if the figure is to 
be reduced, to draw another set of squares to the required scale (Fig. 180), and points in the squares corresponding to those on the other figure can be 
I'eadily marked, and the required figure drawn. The figures shown explain themselves, and need no further remark. 

Note. — Complicated figures are more easily reduced by using the pantograph, an instrument used by the land surveyor for reducing drawings. In every 
case of enlarging, the greatest accuracy is necessary both in drawing, by either of the methods explained, and in manipulating the pantograph, as it is obvious 
that original errors are magnified by enlarging, and new ones are often made. 

119. To construct a Figure similar to a given Figure but with Twriee its Area. — Let ABCDEFGr (Fig. 181) be the given figure. By this time 
you are quite familiar with the fact that areas of similar figures are to each other as the squares on the corresponding sides of the figures. So, obviously, if 
you draw BH at right angles to AB, and equal to it in length, the square on the hypotenuse AH wUl have twice the area of the square on AB (Prob. 92). 
So with A as centre, and radius AH, the out AB produced in 6, then A6, wiU be the base of the new figure. The required figure Abcdefg can now be 
drawn as in Problem 115. 

Note. — Problem 97 will help you to deal with figures whose areas are in any other ratio. 






2 3 4 5 




^^^ 




-4 


^^^fc 


^ 


^^^m 


%^ 


^ 


^^^P 


^ 


^^m 






Figs. 179, 180. — Reduction by use of squares. 



Fig. 181, — Doubling area of figure. 



Fig. 182. 



Fig. 183. 



120. To construct a Figure similar to a given Figure, ABODE, Fig. 182, and having an Area equal to that of given Figure, M, Fig. 183.— 
First reduce the given figures into triangles of equal area (Prob. 99), and then the triangles into squares of equal areas (Probs. 90 and 98). Then set off 
BF and BG. the sides of the squares, along a line BG through B, making any suitable angle ivith AB (this angle in the figure is 90°) ; the side of the 
square representing the figTire ABODE being BF, and that representing the figiu-e M being BG. Join F to A, and through G draw Ga parallel to FA, 
cutting BA produced in a. Then dS> is the base of the required figure, which can be completed by drawing ae, etZ, and dc parallel to AE, ED, and DC 
respectively, cutting the lines BE, BD, and BC produced in e, A, and c. (Refer to Prob. 115.) 

Pkoof. — Area aBcde <£& ,„ , .„. 
: -^^2 (Prob. 93) 

area of M 



But 



area ABODE 
oB^ _BG2 
AB2 ~ BF2 



area of ABODE 



Note. — This represents a very important type of problem, and when you understand the expedients employed, you should experience no difficulty in working 
anv variation of it. 



60 



INDUSTRIAL DRAWING AND GEOMETRY 



121. The Mass-centre, or Centre of Area or Gravity. — The mass-centre of every straig-ht line is its geometrical centre, and the mass-centre of 
any triangle is in a line bisecting it and its base, the distance of the mass-centre or e.g. (centre of gravity) being one-third the length of the bisector from 
the base. 

This can be easily understood by reference to the figure ABC (Fig. 184). We may suppose that the triangle consists of a nximber of lines placed side 
by side parallel to AB. Then, as the mass-centre of each line is its geometrical centre, the line CD,' which passes through all these centres, will contain 
the e.g. of the whole figure. 

But for the same reason BE, which bisects AC in E, contains the e.g., therefore the intersection of these two lines BE and CD is the mass-centre of 




A D B 

Pig. 184.— Mass-centre. 





Mass-centre of quadrilateral figure. 



Fig. 186. 



the figure. Proof — Join E to D, then, as the triangles ADE and ABC are obviously similar, ED will be parallel to CB, and the triangles ED(cjf) and 
CB(ca) are also similar. 

Therefore D(cj?) : (c^)C :: ED : CB 
But it wiU be seen that AD : AB : : DE : BC : : 1 : 2 

Or D(c(/) : icg)C : : 1 : 2, that is {cg)D = J CD, or the mass-centre is J the centre line from the base.^ 

122. To find the Mass-centre of any Quadrilateral Figure. — Let ABCD (Fig. 185) be the given figTire. Then join the opposite corners BD 
and AC. The lines intersect in E. Then set off AF from A along AC equal to CE. Draw BP and DP, and by the previous case find the e.g. of the 
triangle BDP, and it can be shown that this point is the e.g. of the given figure.^ 

Alternative Method. — Each diagonal AC and BD divides the figure ABCD into two triangles ; if the c.g.'s of these two pairs of opposite triangles 
be joined, the intersection of the two lines will also contain the e.g. or mass-centre of the figure. 

' This line is called the axis of skew symmetry. 

^ If over the area of a figure we have a distribution of small equal areas with equal masses (or weights), such that for every bit of the area, however small, 
the mass is proportional to the area, then there is a uniform distribution of mass, and the area is said to be evenly covered or loaded. 

' You will be assisted in this reflection by satisfying yourself that the area of the quadrilateral BADF is equal to the area of the triangle BDC. 



REDUCING AND ENLARGING FIGURES, ETC. 



61 



DRAWING EXERCISES. 

Note. — In working the following exercises it will sometimes be convenient to prick ofi corners of the figures (or suitable points iu them) by placing the page 
of the book over your drawing paper. 

1. Reduce the irregular polygon (Pig. 186) to a similar one, all the linear dimensions being halved. Work this exercise in two different ways, and compare 
the reduced copies. 




























' 


^ 


-V, 














. 






■v. 


) — 






\ 










/ 






; 


\ 






/ 










/ 








s 








/ 


1 








/ 








/ 








/ 










s 
















A 








B 










Fig. 187. 



Fig. 188. 



Pig. 189. 



2. Draw a figure on base CB (Fig. 187), similar to the one on base AB. 

3. Carefully copy the figure given in Pig. 188, and by using squares, as in the figure, draw an enlarged copy, doubling all the linear dimensions. 

4. On ab (Fig. 189) as base, construct an irregular heptagon similar to the one ABCDEF on base AB. And draw two squares whose areas are in the same 
proportion as the areas of the polygons. Further, from these estimate what the ratio of the areas of the polygons is. 

5. Set out the polygon (Fig. 190) to the dimensions given, then draw a similar one but with double its area. 






Fig. 190. 



Fig.: 191. 



Pig. 192. 



6. Draw a figure similar in shape to the polygon in Fig. 186, but with an area equal to that of ABCDEF, Pig. 189. 

7. AB (Fig. 191) is the base of the plot of ground shown. Set out the plot, scale Vj" to the foot, and draw a reduced copy of the plot on base ab which is half AB. 

8. Draw a scalene triangle with sides, 2", 4", and ih", and determine the position of its centre of gravity, or mass-centre. 

9. In two different ways find the mass-centre of the quadrilateral figure (Pig. 192). 



CHAPTER XI 



SYMMETRY AND SYMMETRICAL FIGURES 

123. Introduction. — In Chapter II. you will remember dealing with some simple symmetrical figures. For instance, Fig. 27 shows a 
rectano-ular figure symmetrical about a centre line AB, and this line is called the axis of symmetry, whilst in Fig. 28 we have a square 
fio-ure symmetrical about two axes of symmetry at right angles. You may have often seen pieces of paper folded, and cut with the 
scissors, which when opened out present very pleasing symmetrical figures. Fig. 193 shows a piece of foolscap paper, and MN is the 
folded edge. The folded sheet is cut along the lines MCN to form a triangle, and when the paper is opened out the kite, MCNC2, is 
formed, and MN is the axis of symmetry. If you fold a sheet of, say foolscajD, paper twice, so that the folds or creases are at right 




Pig. 193. Fig. 194. — Axis of symmetry. 





Pig. 195 



Axes of symmetry. 



Pig. 197. — Centre of symmetry. 



angles, and cut it along the original edge of the paper to form a square, ABFC, Fig. 195, then any pattern may be cut, such as 
shown in the figure, and when unfolded you will have made a figure with two axes of symmetefy, as in Fig. 196. If the corner at A 
had been clipped off as shown, then a square hole would appear in the unfolded figure. When once you have seen how easy it is to 
produce symmetrical figures in this way, you will often be fempted to experiment with scissors and paper. 

124. Centre of Symmetry. — If you draw a parallelogram. Fig. 197, and through C, the intersection of its diagonals, you draw a 
number of straight lines, such as MN and OP, across the figure, you will find that they are all bisected at the centre C, which is then 
said to be the centre of symmetry ; and the parallelogram is said to be sjrmmetrical about the point C. 

62 



SYMMETRY AND SYMMETRICAL FIGURES 



^ -sf/ri^ 


L^ 


^^u 


~N 


4^Q 


r^ 


T^^1 


' \ 


-41- -^ 


I ^ 


^y 


\^^ 


n ^y 



c 



% 



125. Use of Squared Paper in Drawing Symmetrical Figures.— Squared paper lends itself to the easy construction of symmetrical 
figures. We have an example in Fig. 198. Suppose the curved figure at the left of the centre line MN be first drawn, the right- 
hand half could obviously be easily drawn by using the corners of the squares as centres of the arcs. Indeed, the positions of any 
points corresponding to others on the other half of the figure can easily be located in this way. 

126. Practical Applications of Symmetrical Figures — Engraving, Lithography, and Printing. — Obviously, the figures E and 3, 
Fig. 199, are symmetrical about the axis of symmetry, MK Xow, suppose that the E is drawn in ink on a sheet of paper, CDEF, 
and the ink remains wet ; if the paper be folded along the line MN", the impression 3 on MNDE will be made ; and whilst this is 
wet you could get an impression of the E on another sheet of paper, 

which would be of course the original E- Now, if you understand 

this simple operation, you know something about the principle made 

use of by engravers and lithographers. For the object of the arts 

practised by these workers is to form on the surface of a block or 

plate of some suitable substance, such as wood, metal, or stone, 

certain figures, of which the impression is printed or transferred 

exactly to some other surface. Now, let us suppose that the E on 

CNMF is the figure reqiured to be printed. Then, obviously, the 

figure on the block must be drawn in a reversed position, as on 

NDEM. For this reason the types used by the compositors to print puj. 198.— Axis of symmetry. 

this page are cast reversed, and placed in the frame by proceeding 

from right to left, in order that, when applied to the paper, the letters they produce may be in their natural position, and be read 

from left to right. 

Thus mere impression does not produce copies equal to, or like the figure on the block or plate, but symmetrical reversed ones. 
Compare an Indiarubber stamp with the impression it makes, or some written matter with its impression on the blotting-pad, and 
you will better understand this. 

127. Stereotyping. — In stereotyping, matrices are engraved, drawn or composed, and by means of them impressions are made on 
plates, which are again employed in the ordinary way to print drawings, music, or writing, etc. Of course, at the first impression, 
the figiires pass from the left to the right (as in Fig. 199), and at the second repass from the right to the left. Therefore in stereo- 
typing, the printed figures are identical on the primitive matrix and on the copies taken from the intermediate plate. 



Fig. 199. — Symmetry by impression. 



EXERCISES. 
Obai. Exbecises. 

1. When has a figure a line or axis of symmetry ? Mention any geometrical figure you can think of that has an axis of symmetry. 

2. Jlention two geometrical figures each of which has two axes of symmetry at right angles to each other. 

3. Give the name of any figure that has a centre of symmetry, 
i. Has a scalene triangle an axis of symmetry ? 



64, INDUSTRIAL DRAWING AND GEOMETRY 

Drawing Exbhcises. 

5. Draw the following figures and mark on them their axes of symmetry : an isosceles triangle, a rectangle, a pentagon, and a hexagon. How many axes of 
symmetry has the last-named ? 

6. Show by drawings the symmetrical figures that can be produced by cutting folded sheets into the following shapes : (a) a scalene triangle with its 
hypotenuse along the crease ; (b) a sector with an angle of 60°, a side along the crease ; (c) a rhombus with a side along the crease. 

7. By using a piece of tracing paper, trace Fig. 187, and reproduce the figure on your drawing paper by pricking it ofi. Then draw the symmetrical figure 
produced by using AB as the line of symmetry. .„.,„„,„ , ,, , ,, . . r ., • ^ 

8. Draw on a piece of squared paper the letter L, assume any axis of symmetry, as m Pig. 199, and then draw the shape the impression of this figure would 
make if it was drawn in ink, and you in blotting it made an impression on the blotting paper. 



CHAPTER XII 



THE ELLIPSE 



128. Introduction. — Look at Fig. 203.^ No doubt you have often heard figures that shape called ovals, although the proper geo- 
metrical name of the figure is ellipse. Strictly speaking, an oval is broader at one end than at the other — in fact, egg-like in form.^ If 
you saw off a piece of broomstick, and the saw-cut is on the slant, as in Fig. 202, you know that the shape of the cut will not be 
circular, but elliptical, as in Fig. 203 ; and you no doubt also know that if a cone is cut, as in Figs. 200, 201, the cut or section is 
also an ellipse.^ In fact, an ellipse may be defined as a curved figure formed by the intersection of a plane and a cone or of a plane 









Fig. 200.— Cone cut 
by plane. 



Fig. 201.— Elliptical 
section. 



Fig. 202.— Cylinder cut 
by plane. 



Fig. 203. — Showing axes 
and a diameter. 



Fig. 204. 



Pig. 205. 



and a cylinder, when the plane passes obliquely through opjmsite sides of either of the solids. 

An ellipse is symmetrical about two lines or axes which are at right angles to each other (Fig. 203). One, the major or 
transverse axis (Fig. 205), passes through the centre, and is the longest line that can be drawn on the figure, the other, the minor or 
conjugate axis (Fig. 206), bisects the major axis perpendicularly, and is the shortest line that can be drawn on the figure. 

' Oval from ovum " an egg." Elliptical mirrors are commonly, but erroneously, referred to as being oval. 

- The projection of a circle on a plane inclined to its surface is also an ellipse. It may interest you later to know that if a triangle move in such a way that 
two of its corners always lie in two fixed lines, the locus of the third corner will be an ellipse. 

You may have been told that the paths in which the planets move round the sun are ellipses, the centre of the sun being one of their foci. Over 3000 years 
passed in studying astronomy and geometry before this beautiful truth was discovered. 

65 J. 



66 



INDUSTRIAL DRAWING AND GEOMETRY 



129. Some Definitions and Properties of the Curve. — If you are reading this book for the first time, you need not trouble about the 
following definitions, etc. They a^Dpear here primarily for reference purposes. 

Diameter. — Any line passing through the centre of an ellipse and terminated both ways by the curve is called a diameter. Hence 
it is manifest that the centre bisects all diameters (Fig. 203). 

Vertices. — The extremities of a diameter are called vertices (Fig. 204). 

Transverse Axis. — The diameter which passes through the foci is called the transverse axis. This is also called the major axis 
(Fig. 205), as we have seen. 

Conjugate Axis. — The diameter which is perpendicular to the transverse axis is called the conjugate axis. This, as we have 







Pig. 206. 



Pig. 207.— Showing the focal distances. 



Pig. 208. 



Pig. 209. 



seen, is also called the minor axis (Fig. 206). Any two diameters are said to be conjugate when the tangents at the vertices of one 
diameter are parallel to the other diameter. 

The Focal Distance. — The distance of a focus from the nearest vertex is called the focal distance, and the distances of the foci ^ 
from any point in the curve the /oca/ distances^ or radii vectors, such as V and v in Fig. 207. 

Tangent. — The line which passes through a point in an ellipse (Fig. 208) and bisects the exterior angle formed l)y the focal lines 
at that jjoint, is called a tangent. 

Normal.^ — A line perpendicular to a tangent at the point of contact (Fig. 209) is called a normal to the curve. This line, there- 
fore, bisects the focal angle. 

Conjugate Diameter. — A diameter which is parallel to a tangent at a given point is said to be conjugate to the diameter which 

passes through this point (the sum of the squares on conjugate diameters is constant). 

TT 11 

The Area of an Ellipse is found by multiplying the product of the major and minor axes by (= 07854, or, say, zr-i very nearly). 

' Foci, plural of focus. The sum of the focal distances is a constant quantity. Due to this property of the curve, elliptical wheels can be geared to give a slow 
forward and quick return motion, each axis passing through a focus, and the distance between the axes or centres being equal to the major axis or the sum of the 
focal distances. 

' Lat. nmnia, " a square or rule ; " perpendicular. 



THE ELLIPSE 



67 



Simple Problems relating to Ellipses. 



130. To draw aa Ellipse (First Method) as the Locus of a Point. — The elliptical curve may be generated by a point moving in 

such a way that its distance from a fixed point (called a focus), Fig. 210, is in a constant ratio to its perpendicular distance from 

a fi.xed straight line (called a directrix), the generating point being nearer to the fixed point than to the line. Every complete 

ellipse has two foci and two directrices, as shown in the figure. Let A, E, D, and B be points in the curve; then the ratio 

referred to — 

focus to vertex CA CE CD CB ^ . .^ 

gjj = gji = eccentricity. 



vertex to directrix AF EG 



If this be understood, you will be able to easily determine the positions of any number of points in the curve when this ratio (or 
eccentricity, as it is called) is known. And a fair or flowing line through the points so determined will give the curve. 




Fig. 



210. — Ellipse, as the locus of a 
point. 





FiQ. 



211. — Ellipse, constructed by 
concentric circles. 



Fig. 212.- 



-Ellipse, drawn by string 
method. 



131. To draw an Ellipse, having given the Major and Minor Axes— Second Method: By Conoentrie Circles. — Let AB and CD, Fig-. 211, 
be the two given axes. Place them at rig-ht angles to each other at their centres O. Then with O as centre, and radii OA and OD, describe two circles 
(the major and minor auxiliary circles), and from centre draw Knes ("at any angles to AB) cutting the circles, as at GE and HP. Prom E and P draw 
lines EI and PJ parallel to CD (the minor axis), and through G and H draw parallels to AB (the major axis), cutting the parallels to CD in I and J, 
two points iu the required curve ; so draw a fan Hne through AIJD, and a quadrant of the eUipse is formed. Complete the curve, by finding other points 
La the same way, as shown in the figure, or by symmetry. 

132. To draw an Ellipse, having given the Major and Minor Axes — Third Method: Mechanically, by means of a Piece of String and 
Two Pins. — Let AB and CD (Pig. 212) be the given axes. Place them at right angles to each other at their centres O. Then with radius AO (half 
major axis), and centre C or D, describe arcs cutting AB in P and P.^. Then these points are the foci of the ellipse. In each of these points 
stick a small pin ; also place one in C (or D). Then pass a thread or string I'ound the three pins, and tie the ends, making the string taut. The 



68 



INDUSTRIAL DRAWING AND GEOMETRY 



string- now foi-ms a triangle, PCrj. Substitute a pencil for the pin at C, and move it along, keeping the string taut, and the pencil wiU trace a 
true ellipse.' 

Note. — This method of drawing an ellipse is of great service to practical men, as the curve can be readily dravpn on the work or material, but great care must 
be taken not to vary the tension of the string. Gardeners trace their elliptical flower beis in this way, stakes or poles being stuck in the ground at F and F^, 
around which a cord is used. 

133. To draw an Ellipse, having given the Major and Minor Axes— Fourth Method: By Paper Trammel. — Let AB and CD, Fig. 213, be 

the two given axes at right angles to each other at their centres O. Then if on a strip of paper, or the straight-edge of a card, the semi-axes be marked 
from any point, the strip can be used to find points in the curve in the following way. The strip ac, shown in the figure, is marked with ac equal to AO, 
the semi-major axis, and the distance be equal to 00, the semi-minor axis. Then, if the points a and 6 on the straight-edge are kept on the axes, as 





Fig. 213. — Ellipse, paper trammel method. 



■'/////////////,WM^///////.^o 

Pig. 214. — Quadrant of an ellipse as 
locus of a point in the line AB. 




Pig. 215. — Rectangle and the 
inscribed ellipse. 



shown, the point c will trace the curve. When the two axes are nearly the same length, ac and he should be set off each side of c, as shown at &' 
and a' on the second strip. The curve is then traced by c', whilst a! and 6' are kept on the axes and the axes produced. Tracing cloth or paper can be 
used with advantage ; any line on the cloth or paper can then be used instead of one of its edges, and the points pricked through c or c'. 

Notes. — 1. This is a favourite expedient with draughtsmen to rapidly find a few points in an ellipse. 

2. The carpenter's elliptical trammel is based upon this principle. 

8. It directly follows from this problem that if we cause a line AB (Pig. 214) to move with its ends A and B on two lines AD and BD at right angles 
to each other, any point C on the line will trace the quadrant of an ellipse ICH. It is obvious that when the point is at E, in the middle of AB, the curve traced 
will be a quadrant of a circle, GEP. 



' You win understand this expedient when you remember that the sum of the focal distances of any point on the curve is always equal to the major axis. 

(Pig. 212) PD -f PjD = AB. This is important, and should not be forgotten. 



Thus 



THE ELLIPSE 



69 



134. To draw an Ellipse, having given the Major and Minor Axes — Fifth Method : By Radial Lines from the Ends of the Minor 
Axis. — Let AB and CD (Fig. 215) be the g-iven axes at right angles to each other at their centres O. Then thi-oug-h AjB and CD draw the rectangle 
EFHG-. and divide AO and AG- into any suitable number of equal parts (say six) in the points 1, 2, 3, 4, 5 and 1', 2', 3', 4', 5' respectively. From D 
di'aw lines to pass through the points 1', 2', 3', 4', 5', and from C draw lines passing through the points 1, 2, 3, 4, 5 to cut the other radial lines in the 
X)oints a, b. c, d, e, which are points in a quadrant of the required eUipse. A repetition of this construction for the other quadrants completes the figure, 
or the figure may be completed by symmeti-y. 

Note. — This construction is also applicable to cases where conjugate diameters do not intersect at right angles as they do above (see Prob. 188). 

135. To determine the Major and Minor Axes of a gip^en Ellipse. — Commence by drawing any two parallel lines across the given figui-e, such 
as EF and GH (Fig. 216). Bisect each of them at L and M respectively. Join L and M, and produce the line in both directions to cut the figure in 
I and J. Then IJ is a diameter/ and its centi-e 0, which is found by bisection, is the centre of the ellipse. "With this point O as centre, describe any 
arc cutting- the curve in Q and P ; and bisect the arc QP in R. Join OR, and produce the hue both ways to cut the curve in A and B. Then AB is 
the required major axis, and the minor axis CD is found by drawing a perpendicular to AB at its centre O, cutting the curve in C and D. 

136. To draw a Tangent to a given Ellipse at a Fixed Point in the Curve. — First find the major axis AB (Fig. 217), as in the previous 




G C 

Pig. 216. — Construction giving the 
major and minor axes. 





Pig. 217. — Tangent and normal at fixed 
point in the ellipse. 



Pig. 218.- 



-Ellipse, given conjugate 
diameters. 



problem, and the foci F and Fo, as in Prob. 132. Then join the fixed point P to F and Po. Produce P.,P to G, and bisect the angle GPF by IJ. wliich 
is the required tangent. 

A very important property of the ellipse is that the focal lines FP and FjP make the same angle at P with the curve (or with its 
tangent, IJ). 

Note. — As you are probably aware, if a ray of light impinge on a mirror, or a wave of sound on a flat surface, the light or sound is reflected ; and the angle of 
reflection is equal to the angle of incidence. The same law holds good for curved surfaces ; thus, if we had a source of light at P (Pig. 217), and a ray impinged on 
au elliptical mirror APB at P, the angle of incidence would be FPI, and that of reflection P„PJ. It follows from this fact that all the rays from P would be 
reflected to the other focus P„, which would become a luminous point. The same would occur with sound: if the ellipse represent the plan of a building, a 
whisper at either focus P or P„ would be heard at the other, although perhaps it could not he heard at any other part of the buUding. In some foreign prisons 
a cruel use has been made of this echoing property of the curve. 



It wUl be noticed that Unes EP and GH are double ordinates of the diameter IJ, as the lines OD and 00 are ordinates o£ the axis AB. 



70 



INDUSTRIAL DRAWING AND GEOMETRY 



137. To draw a Normal or Perpendicular to a given Ellipse at a Fixed Point in the Curve ^ (Pig. 217). — First find the major axis AB 
(Prob. 13.5) and tlie foci F and F,, as in Prob. 132. Then join the given point P to F and F^. Produce F^P to G, and FP to H. Bisect the angle 
GPH by PK, which is the required normal or perpendicular. 

138. To draw an Ellipse, when two Conjugate Diameters, other than the Major and Minor Axes, are given intersecting each other 
at their Centres — First Method: By Radial Lines from the Ends of One of the Axes. — Let AB and CD (Fig. 218) be the given con- 
jugate axes. Then through the ends of AB and CD draw the parallelogram EFHG, with sides parallel to the given axes. Then the 
points a, b, c, d, e in the quadrant AD of the figure may be found in the same way as in Prob. 134, and the explanation given equally 
applies to this figure. 

139. Second Method : By" using the Auxiliary Circle. — Let AB and CD (Fig. 219) be the given conjugate diameters intersecting in 




\ 


c 


/tl\\ \ 
/ \ \ 


o / y\ 


\ W 


/m K \ 




Fig. 219.- 



-Ellipse, given conjugate 
diameters. 



Pig. 220. — Ellipse, by circular arcs. 



Pig. 221. — Ellipse, by circular arcs. Pig. 222. — Ellipse, by circular arcs. 



0. Then with as centre describe the auxiliary semicircle AO'B, and divide AB into any number of parts in a, i, c, d, e,f, and from 
these points, and at 0, erect perpendiculars to AB to cut the semicircle in a, b', c', d', e',f,' and 0'. Next, join 0' to C, and through 
the points a, b, c, d, e, f draw lines parallel to the diameter CD, intersecting lines through a', b', c', d', «', /' parallel to O'C in 1, 2, 3, 
4, 5, and 6, which are points in the required ellipse. The other half of the curve can be drawn in the same way, or be put in by 
symmetry. 

140. Approximate Method of constructing Ellipses hy Circular Arcs, having given the Number of Centres- — First Method (By Three 
Centres) : When the Major and Minor Axes are given. — Let AB and CD (Fig. 220) be the major and minor axes perpendicular to each 
other at their centres 0. Then through C and A draw parallels to AB and CD, intersecting in E. Bisect AE in F, and join CF and 
ED, intersecting in G, which is obviously a point on the true curve (Prob. 134). Next bisect CG in H by the perpendicular HJ, 
cutting CD produced in J, which is the centre of curvature at C. With J as centre, radius JC, describe the arc CKI, cutting a line 

' This is a problem that sometimes occurs in connection with an elliptical arch when the intradoa and extrados are parallel. When all the normals to an 
ellipse are intersected by a curve at equal distances, the carve is said to be parallel to the ellipse. This curve will not, as the student might suppose, be an ellipse, 
although in most cases it will much resemble one; it will have in every case a different character. A circle parallel to a circle is a concentric circle; with this 
exception, all parallel curves are different in character. 



THE ELLIPSE 71 

parallel to AB through J in I. Join I to B, and produce it to cut the arc in K ; and join K to J, cutting AB iu M ; then with centre 
M, radius MK or MB, describe the arc KB. Then J and M are the centres of curvature of the ellipse at C and B, and the two arcs 
CK and KB form a quadrant of the required ellipse and give a close approximation to the true curve. Make AN = BM, and through 
J and iSI" draw JL. Then the semi-ellipse ACB is formed of the three arcs AL, LK, and KB, described from the centres IST, J, and M 
respectively. And the other half of the figure can be completed by symmetry. 

141. Second Method (Another Method by Three Centres) : When the Major and Minor Axes are given.^Let ABand CD (Fig. 221) be the 
given axes mtersectini? in 0. Then make AE on AB equal to the minor axis CD. and divide BE into three equal parts in F and G. With as centre, 
radius FB (two of the parts), mark ofl H and J. Use H as centre, radius HJ, and cut CD produced in M. Then with M as centre, radius MC, describe 
the arc CN, cutting a line through M and H in N. With H as centre, radius HN or HB, describe the arc NB, which completes the quacb-ant CNB 
of the required ellipse. The figure can be completed by symmetry. 

142. Third Method (By Three Centres) : When the Minor Axis is not less than Two-thirds the Major Axis.— Let AB and CD (Fig. 222) 
be the given axes intersecting in O. Then with O as centi'e, radius OB, describe an arc BE, cutting OC produced in E. Then on OB construct the 
equilateral triangle OBJ ; join J to E, and thi-ough C draw CF parallel to EJ, cutting BJ in F. Through P draw FG parallel to OJ. cutting OD or 
OD produced in G and AB in H. Then G and H are the requii-ed centres, and the radii are HF and GF respectively. With these centres the arcs can 
be described as shown, and the ellipse completed by symmetry. It will be noticed that the angle subtended by each of the three arcs is 60°. 

Note.— This construction gives a good curve for an arch, as it can easily be set out, is pleasing to the eye, and gives a very suitable waterway. When the 
minor axis is less than two-thirds and greater than half the major axis, f3ve or more centres should be used, as three do not give a curve of agreeable form. (See 
Author's " Geometrical Drawing," p. 132.) These curves are sometimes called the basket-handle arches, or curves of many centres. 



EXERCISES. 

Typical Oead Exercises. 

1. What is the difference between an ellipse and an oval ? 

2. How many axes of symmetry has an oval ? 

3. The end of your round pencil is a circle. If you cut the end in a slant direction, what is the shape of the cut surface or section ? 

4. What is the name of the longest line you can draw across an ellipse ? And what the shortest line ? 

5. If you draw a line that just touches the curve of an ellipse without cutting it, what is the name of the line in relation to the curve ? 

6. What is a normal to an ellipse ? 

7. Mention any object or part of a structure that is elliptical in form. 

8. In what shaped paths do the planets move round the sun ? 

Drawing Exercises. 

9. In a rectangle SJ" x 2" inscribe an ellipse. 

10. Construct an ellipse with axes 3" and If" by using a strip of stiff paper to find points in the curve. 

11. The major and minor axes of an ellipse are 3i" and 2" long respectively. Draw the curve by three different methods, and find its foci. 

12. After drawing the ellipse in the previous problem, show how you would determine its axes, supposing that their positions are unknown. 

13. Draw a tangent to an ellipse whose axes are 3" and li", the point of contact to be IJ" from an extremity of either axis. 

14. Draw a semi-ellipse (axes of 4" and 2"), and set out a number of normals to the curve 0-75" long (externally), and through their ends draw a curve 
parallel to the ellipse. 

15. The distance between the foci of an ellipse is 2i", and the major axis is SJ" long. Draw the ellipse. 



72 INDUSTRIAL DRAWING AND GEOMETRY 

16. Two conjugate diameters of an ellipse are 3" and 2i" long, and they are inclined to one another at an angle of Y0°. Set out the ellipse in two 
different ways. 

lY. The diagonals of a rhomhus are 4" and 21". Draw the figure, and inscribe in it an ellipse. 

18. A rectangle, 3" x If", circumscribes an ellipse. What is the arithmetical diiJerence between the areas of the ellipse and the rectangle ? 

Note. — The area of an ellipse is equal to the product of the axes multiplied by - . 

19. Draw a semi-ellipse (axes 5" and 2i"), making the major axis 5" and the semi-minor axis li" ; then draw a parallel curve (inside) i" from it. 

Note. — This is a problem that sometimes occurs in connection with an elliptical arch when the intrados and the extrados are parallel. When a number of 
equal normals to an ellipse have their ends in a curve, the curve is said to be parallel to the ellipse, as we have seen. 

20. Draw an ellipse with axes 2f " and If", and find the centres of curvature of the figure at the extremities of the major and minor axis ; also of one other 
point in the curve. 

21. Draw in two different ways an approximate ellipse by means of circular arcs, using three centres and making the axes 3" and 2". 



CHAPTER XIII 



THE PARABOLA 



143. A Parabola^ may be defined as a curved figure formed by the intersection of a plane and a cone, when the plane passes through 
the cone parallel to its side,'-^ as in Figs. '223, 224; or, in other words, the curve is a conic section, or conic, as it is sometimes called. 
In the language of co-ordinate geometry, a parabola is the locus of a point which moves so that its distance from a fixed point called 
the focus is equal to its distance from a fixed straight line called the directrix, as we shall see later. 

The parabola is one of the most important curves the practical man has to deal with, as by its use many interesting problems 
can be graphically solved. The curve can be drawn 
from a great variety of data, but only a few simple 
cases come within the province of this work. 

144. Definitions and Properties, etc. — If you are 
reading this chapter for the first time, don't trouble 
about these definitions, etc. They are here mainly for 
reference purposes. 

Diameter. — A straight line perpendicular to the 
directri.x; (Fig. 225), terminated at one extremity by 
the parabola and produced indefinitely within, is called Fig. 223.— Gone Fig. 
a diameter. 







I' 


















^^ 


Diameter 


\ 






c 


/ 




J^ 


P.[ycrlm 


Axi':- 














•<; 


V 






Fig. 223.— Gone 
out by plane. 



224.— Parabolic 
section. 



Fig. 225. 



Fig. 226. 



A diameter of a curve may be defined as the locus of the middle points of a series of parallel chords. 
Axis. — The diameter of a parabola which passes through its focus is called its axis (see Figs. 225 and 235). 
Vertex. — The point in which a diameter meets the parabola is called its vertex (see Fig. 225). 
Principal Vertex. — The vertex of the axis is called the principal vertex (Fig. 225). 

Ordinate. — A straight line bisected by a diameter and terminated both ways by the parabola (Fig. 226), is a douUe ordinate 
of that diameter. An ordinate x is proportional to the square root of its abscissa h. 

' The path of a body that is thrown obliquely in a vacuum is a parabola. A shell fired from a mortar moves approximately in a parabolic path. The orbits 
described by comets appear to be parabolas, the sun being at the focus. But they are known to be ellipses very much elongated. 

= Or parallel to a single generator is, perhaps, a better way of defining it. The sides of a parabola come closer together as the cutting-plane approaches the 
side of the cone, so the limit of the figure is a straight line. 

73 



■74 



INDUSTRIAL DRAWING AND GEOMETRY 



Abscissa. — The segment of a diameter between its vertex and an ordinate is called an abscissa (see Fig. 226). 
Chord. — A straight line cutting the parabola in two points is called a chord. 

Tangent. — The tangent at any point P in a parabola ( Fig. 227) bisects the angle FPE between the focal distance EP and PE, a 
perpendicular on the directrix CE. 

Note. — From any external point two tangents can be drawn to a parabola. Tangents wMoh meet in the directrix are at right angles to each other. 
Sub-tangent. — The distance JG (Fig. 227) between the intersections on the axis of the tangent at a point P and a perpendicular 
to the axis from P is called the sub-tangent at P. 

Note. — The extremities J and G of the sub-tangent are at equal distances from the vertex. If this be remembered, a tangent at any point P can be directly 
drawn by first finding the sub-tangent. 

M 




Fig. 227. — Tangents and normals to a parabola. 





Fig. 228.— Parabolic reflector. 



Fig. 229.— Paraboloid. 



Normal. — The normal PH (Fig. 227) to a tangent is a line at right angles to the tangent at the point of contact. It bisects the 
exterior angle EPK, formed by. the focal distance and a perpendicular on the directrix CE. 

Sub- normal. — The distance GH (Fig. 227) between the intersections on the axis of the normal at a point in the curve and a 
perpendicular from the point to the axis is called the sub-normal. It is a property of the parabola that the sub-normal is constant. 
The sub-normal equals twice the distance of the focus from the vertex. That is to say, it equals the distance between the focus and 
directrix. 

145. Diameter and Focal Line. — Any diameter, such as EP produced (Fig. 227), is inclined at the same angle ^ to the curve 

as the focal line FP. 

' This is a valuable property of the parabola. For if we imagine the curve MEVN (Pig. 228) to be the plan of a parabolic wall capable of reflecting light, 
and that we have a source of light at the focus F, it means that parallel rays AD, BE, OF will be reflected ; the angle of reflection being equal to the angle of 
incidence {refer to note on Prob. 136). Now, if we revolve the figure about its axis VW, a surface of revolution will be formed (in a geometrical sense), called a 
paraboloid. Indeed, we have in some lighthouses a plated copper mirror, having the form of the paraboloid (Fig. 229), and all the rays reflected from the internal 
surface when the source of light is at the focus F (Fig. 228) form a beam of light having the circular end for its base. You no doubt have often seen such a beam 
of light-on the sea coast, or at sea. Sometimes, it is made to revolve, that it may be seen from all quarters of the compass. 



THE PARABOLA 



75 



146. To draw a Parabola when its Axis and Base are given— First Method.— Let AB (Fig. 230) be the base, and CD the axis or 
height. Then draw CD perpendicular to AB at its point of bisection, and through D draw EF parallel to AB, and intersecting the 
perpendiculars through A and B in E and F respectively, completing the parallelogram ABFE. Divide AC and CB into any 
number of equal parts (four are taken), and through these divisions draw parallels to CD. Next divide AE and BF into the same 
number of equal parts (four), and through these divisions draw lines to D. These lines will intersect the parallels to CD, giving 
points a, h, c, etc., in the required curve, as shown. 

Notes.— 1. This construction depands upon tlie fact that if a diameter be drawn through tlie centre point of any chord, the tangents at the ends of the chord 
intersect on the diameter, and the carve cuts tlie diameter at the centre point between the intersection of tlie tangents and the chord. Thus, BD is a chord ; and the 



E a b c D 



Distributed Load W 




A 3 2 I C 



2 3 B 




Fig. 230. — Parabola on base AB. 



L 



Fig. 231.— Loaded cantilever. 




3 Z 

Pig. 232. — Bending moment diagram. 




C 8 

Fig. 283. — Parabola on base AB. 



diameter through G will intersect it in its centre, J. DF is the tangent at D, and the tangent at B will also pass through G (refer to note on the Sub-tangent) ; 
and H, which bisects GJ, will be a point in the curve, since (by similar triangles) GJ : JH : : F2 : B2. 

2. This is the curve (ADB) which represents the variation of bending moment in a beam suppoiting a uniform load. AB would represent the length of 

the beam, and the height CD the greatest bending moment — g— , whilst the abscissae from the divisions in AB measure the varying bending moment ; for the 

o 

distance of any point in this curve from EF varies as the square of its distance from CD, and all curves which satisfy this condition are parabolas. 

WL 



In the case of the cantilever (Fig. 231), supporting a distributed load W, the greatest bending moment is, as you perhaps know, 



And if DA (Pig 232), 



is made to represent this, the variation of the bending moment from the free end E to the fixed end D, is represented by the parabolic curve EafccA, the 
construction of which should now be obvious. 

147. To draw a Parabola when its Axis and Base are given — Second Method (by Tangents). — Let AB (Fig. 23-3) be the base, 
and CD the axis, as in the first case. Then set up CD perpendicular to AB at its centre point C, and produce CD to E, n aking 
DE equal to CD. Join AE and BE, and these two lines will be tangents to the required curve (refer to note on the Sub-tangent). 
Now, if these lines be each divided into any number of equal parts (sis in the figure), and the divisions be numbered and joined as 



76 



INDUSTRIAL DRAWING AND GEOMETRY 



shown, the lines 1 1', 2 2', etc., are also tangents, and it is only necessary to take a sufficient number of divisions for an almost 
perfect curve to be formed with very little touching up by hand. 

148. To describe a Parabola when its Base, Height, and Inclined Axis (which passes through the Centre of the Base) are given. — 
In this case it is required to have the greatest height at some point not directly over the centre of the base. The position of the 
highest point, D (Fig. 234), in relation to the base fixes the inclination of an axis, CD, which bisects the base. If AB be the base, 
and CD this axis, the curve can at once be drawn by first drawing the circumscribing parallelogram, ABFE, and by finding points 
in its sides, as shown in the figure, and explained in Prob. 146. Of course, if desirable, points can be interpolated between any two 
points already found by subdividing the corresponding spaces on AE and DE. 

Note. — It will be seen in the Pig. 234 that the tangent PH at P bisects GD in H. 

149. To describe a Parabola when the Directrix and Focus are given. — Let AB (Fig. 235) be the given directrix, and F the focus. 




A3 2 I C I 2 3 B 

Pig. 234. — Parabola on base AB with axis inclined. 




Pig. 235. — Parabola as locus of a point. 



Then through F draw the axis CFH perpendicular to AB and cutting it in C. Bisect CF in D, and D, being a point on the axis 
equidistant from the focus and directrix, is the vertex. Next draw a number of indefinite lines parallel to the directrix, such 
as LK, JE, and HG, etc. Then, to find points in the curve contained in these lines, begin with line JE. Its distance from the 
directrix AB is CJ ; so with this radius and with centre F cut JE in E, and similarly with HG, take radius CH, centre F, and cut it 
in G. Then both E and G will be points in the curve, as they are ec[uidistant from the focus and directrix. A third point, K, is shown 
in the figure, and as many more as may be required can be found in this way, and the curve drawn through them. 
It will be seen that the curve is the locus of a point equidistant from a given point F, and a given line AB. 



THE PARABOLA 77 

EXERCISES. 

Typical Oeal Exeboises. 

1. In what direction must a cone be cut to give a parabola ? 

2. A segment of a parabola has a base of i" and an axis perpendicular to it 3" long. What is the area of the figure ? 
Note. — The area of a parabola is two-thirds that of the circumscribing rectangle. 

3. Define a diameter of a parabola. 

i. A projectile fired from a gun moves in a curved path till it comes to rest. What is the name of this curve ? 

Dbawikg Exeecises. 

5. The base and height of a parabola are 3" and 2" respectively. Describe the curve in two different ways. 

6. The axis of a parabola is 3" in length, and makes an angle of 65° with its base, which is 2J" in length. Construct the curve. 

7. Set out the parabola in Question 5, and mark a point on the curve 1" from the vertex, and through the point draw a tangent to the curve by two 
different methods. 

8. The base of a parabola is 3J", and its height 2J", the highest point above the base being vertically above a point in the base 1" from the centre. Draw 
the curve. 

9. The base of a parabola is 4", and an axis of the curve passing through the centre of the base is inclined 70° to the base and is li" long. Construct the curve. 

10. Set out the parabola in Exercise 5, and find its focus and directrix. 

11. The focus of a parabola is J" from the directrix. Describe the curve, making the axis 3" long. 

12. The distance between the vertex and focus of a parabola is j". Set out a small part of the curve approximately by means of circular arcs. 
18. The height of a parabola is 2J", and its base 2". Describe the curve by the method of tangents. 



CHAPTER XI y 

THE HYBERBOLA 

150. Introduction. — You will remember that it is explained in the preceding chapters that the ellipse and parabola may be defined 
as curbed figures formed by the intersections of planes and a cone — in the former, when the plane passes through opposite sides of the 
cone, and in the latter, when the cutting plane is ^jaraZZeZ to the side (or parallel to a single generator). In a similar way the 
hyperbola, which is the other important conic, may be defined as a curved figure formed by the intersection of a plane and a cone 
when the plane is parallel to any two generators, such as those whose elevations AB are shown in Fig. 236. All such planes cut 
both sheets of the conical surface, and the curves of such intersections have two branches,^ as they are called, which are unlimited in 
extent. 

As in the cases of the ellipse and parabola, this definition does not bring out the property of the curve which furnishes the most 
convenient method of constructing it. So we must again resort to the language of co-ordinate geometry, and define a hyperbola 
as the locus of a point which moves so that its distance from a fixed point called a focus bears a constant ratio to its distance from 
a fixed straight line called the directrix, the ratio being greater than unity. 

So we have in the parabola a ratio of equality ; in the ellipse, one less than unity ; and in the hyperbola, one greater than unity. 

The eccentricity of the curves is the numerical value of these ratios. The sides of the hyperbola become straighter as the 
cutting plane approaches the axis of the cone, so the limit of the figure is a pair of straight lines when the plane contains the axis. 

The most important features of this interesting curve are shown in Fig. 237. AB is the transverse axis ; EG the conjugate 
axis ; Fi, F2 the foci ; and F3, F4 the foci of the conjugate to the curve. 

Asymptotes. — The diagonals DH and SJ (Fig. 237) of the rectangle, formed by the tangents to the hyperbola and its conjugate 
at their vertices, are called the asymptotes of the hyperbola. They continually approach the curves without limit, but never meet 
them. 

The Rectangular Hyperbola. — When the axes AB and EG (Fig. 237) of an hyperbola are equal, the curve is called equdateral or 
rectangidar, and the angle between the asymptotes is a right angle, and the product of the abscissa QX and ordinate QY of any 
point Q iu the curve is a constant. 

For drawing purposes this is the most important case, and we will confine our attention to it. 

1 Obviously if the cone is generated by a line AC (Fig. 236) revolving about an axis AG, and if the line is produced beyond A to D, an inverted cone, DAE, 
sometimes called the opposite cone, is generated at the same time, with a common apex A. Our cutting plane cuts this other cone also and gives the other 
branch of the hyperbolic curve. Briefly, if a cone be cut into two parts by a plane, which, if continued, would meet the opposite cone, the section is called a 
hyperbola. 

78 



THE HYPERBOLA 



79 



151. Given an Ordinate and Abscissa, EF and ED (Fig. 238), of a Point in a Rectangular Hyperbolic Curve, and the Axes (or 
Asymptotes) AB and AD, to draw the Curve. — Complete the parallelogram ABCD. Next divide EC into any number of parts (prefer- 
ably equal parts) (say five) in 1, 2, 3, etc.; and through these divisions draw lines parallel to DA, terminating in DC and AB. 
Lines may now be drawn through the corner A to pass through the divisions 1, 2, 3, etc., on DC, as shown; then at their intersections 
1', 2', 3', etc., with EF draw lines I'a, 2'b, 3'e, etc., to intersect the lines through 1 1, 2 2, 3 3, etc., in a, b, e, etc. (points in the required 
ciu've), as shown. If these points be now joined by a flowing line, any point x in the curve will be distant i'rom the axes AD and 





A-iixiliary 



circle 




Fig. 236. — Cone out by plane, 
giving hjijerbolio section. 



Pig. 237. — Hyperbola, with its most important 
features. 



F 1 2 3 4 
Fig. 238. — Rectangular hyperbola. 



AB such that the product xy x xz = FE X ED ; for by easy Euclid it can be proved that the rectangles AFED and AlaG, etc., are 
equal ; and therefore the product of the rectangular distances of all points in the curve from the axes is a constant, as we have 
seen it must be. 

Notes. — 1. This curve is of great practical importance, as by it the varying pressure of an expanding gas can be shown in relation to its volume. In other words, 
it is the isothermal curve (curve of equal temperature) used by engineers to show the varying pressure of expanding gases, and the indicator diagrams taken from the 
cylinders of heat-engines approximate more or less to the form of the figure ABcfcED, where AD and FE would represent the initial absolute pressure, Be the 
terminal absolute pressure, and AB the stroke of the piston ; the initial volume of the gas would be represented by AFED, and the terminal volume by ABCD. 

2. Area. — The area of the figure ADEeB (Fig. 238) is often required, and is represented by the mean height of DE6e above AB, times AB, or by (AF x FE) 

+ Uf X PE X log. ^\ 

The log is the hyperbolic one, which is the ordinary or common tabular log multiplied by the modulus 2-30258. 

8. The mean height of D&ae = the mean pressure. If the area of the figure ADEeB be divided by the length AB, the quotient will equal the mean height. 

Example. — Let the scale of pressures be 80 lbs. per sq. inch = 1", and the area = ij", and the length : 



Then the mean height = MH' = 



length 



:4". 

: 4J" = l-js", and therefore mean pressure = 80 x MH' = 80 x IVj" = 85 lbs. per square inch. 



80 INDUSTRIAL DRAWING AND GEOMETRY 

EXERCISES. 

Typical Obal Esekoises. 

1. Explain in what way a cone must be cut to give a hyperbolic section. 

2. When is a hyperbolic curve called rectangular or equilateral ? 

3. Why do engineers sometimes caU the rectangular hyperbola an isothermal curve ? 

Deawing Exebcises. 

4. A point P moves in such a way that the product of its distances from two rectangular axes ' is constant. Trace the point in the direction of one of the 
axes for a distance of 3", placing it at the starting-point li" from that axis, and 1" from the other. Note. — Eefer to Problem 151. 

5. The absolute pressure of a gas is 150 lbs. per sq. inch, and it is expanded tiU its absolute pressure is 15. Show by a diagram how the volume varies with 
the different pressures, assuming that the temperature remains constant. Note. — Kefer to Problem 151. 

6. A volume of 10 cubic feet of air at atmospheric pressure (15 lbs. per square inch) is compressed to a pressure of 75 lbs. per square inch isothermally. Show 
by a diagram how the volume varies with the increasing pressure. Note. — Eefer to Problem 151. 

1 The axes referred to here are the asymptotes of the rectangular hyperbola. When these are at right angles the curve is of course called rectangular, as 
we have seen. 



CHAPTER XV 



SPIRALS AND MISCELLANEOUS CURVES 



152. Introduction.-— If a line rotate in any plane about one of its extremities as a fixed point in that plane, and a point con- 
tinuously travels along the line in the same direction according to some definite law, the curve traced by the point is called a spiral, 
and the fixed point is called its pole. A fixed line in the plane passing through the pole, from which the angle passed through by 
tha moving line (called the position or vectorial angle) can be measured, is called the initial line, and any line joining a point in the 
curve to the pole is called a radius vector. (A spiral may also be defined as the 
path of a point whose radius vector is proportional in some way to its vectorial 
angle.) 

As this radius vector increases in length as the position angle becomes larger, 
and the magnitude of the angle may increase without limit, it is obvious that spirals 
can extend to an infinite distance from the pole, and that the number of convolutions 
equals the position angle divided by 360°. 

153. The Involute of the Circle. — If a perfectly flexible line or inextensible 
thread be wound round any curve so as to coincide with it, and be kept taut as it is 
unwound again, a point in the line will trace another curve called the involute of the 
first curve, this first curve being called the evolute ^ relatively to the involute. 
Now, for example, if the curve around which the thread is wound be a circle, then 
the curve traced will be an involute of the circle. You can easily draw such a curve 
if you take a reel of cotton and press the circular end on the table, as in Fig. 239. 
A pencil attached to the free end of the thread will enable you to trace such a curve 
as Pi P2 . . . P12 on a sheet of paper. The free part of the thread will be a tangent 
to the circle (or evolute) at the point of contact as it leaves it. This point of contact is the instantaneous centre of motion, and the 
centre of curvature at the tracing point. 

154. To describe the Involute of a Circle. — Let ACP (Pig. 239) be a diameter of the given circle, and P a i^oint in the required curve. Then 
di-aw PPii tangent to the cii-cle at P, and make the distance P to P,, equal to the circumference of the circle ( = dia. >- ir). Kext divide both the circle and 

' The evolute of a curve is the envelope of the normals to the curve. 

81 G 




Involute of a circle. 



82 



INDUSTRIAL DRAWING AND GEOMETRY 



the line PPjj into any number (say twelve ') of equal parts in 1', 2', 3', etc., and 1, 2, 3, etc., respectively, and at the points 1', 2', 3', etc., in the circle draw 
tangents. Now, it is ohvious that if P be the fixed end of tlie thread, and P,, the movable end, P,, will cover P when the thread is wound round the 
circle, and that as it unwinds the point P, will be from 11' a distance along- the tangent at 11' equal to PI along PB, and that when P is at P, its distance 
from 10' will be equal to P2 on PB, and so on till all the other points P3, P4, Pr„ etc., have been fixed by taking the corresponding- distances from P along 
PB, and marking- them off along the tangents from the tangent jjoints. A flowing curve through the points Pj, Pg, P3, etc., will give one convolution of 
the required involute of the circle through the j)oint P. 

Note. — It is obvious that the moving string is tangent to the circle and normal to the curve at any point, and that a line at right angles to this line at the 
generating point is a tangent to the curve at that point. Thus P,A is a normal at P„ and MN a tangent at that point. 

154a. Cams. — The term " cam " is applied to a curved plate or curved groove used to communicate motion to another piece by the 
action of its curved part. We have such a plate, P, in Fig. 240 fixed to the shaft S and rotating with it, its edge in contact with 
the roller-end E, of the oscillating lever HR of a shearing machine. When the end E of the lever is raised by the cam the other end 
H, to which one of the shear blades is fixed, descends on K, the piece of metal to be sheared. If you examine the cam P you will 
see that when it rotates in the direction of the arrow the roller E is gradually raised whilst in contact with the part CFD, during 
which the shearing work is done. Then, whilst in contact with the part of the cam BE, the heavy end R of the lever quickly falls 

(about the trunnions or axis A), giving to the upper shear 
blade at H a slow downward and a c[uick upward motion. 
The part EC of the cam is concentric with the shaft S, 
and therefore there is no motion of the lever whilst the 
roller E is in contact with this part ; in fact, an interval 
of rest occurs, during which the piece to be sheared can 
be adjusted. 

155. To set out a Cam so that a Point reciprocated by 
it in a Straight Line will move as follows : whilst the Cam 
is uniformly rotating 180°, uniformly upwards l.V", interval 
of Eest for 30° ; uniformly down ^' for 60° ; and the re- 
maining 1;^" for the 90°. — Draw any straight line AG 
(Fig. 241), and markoif from s, a point in it, any distance 
sa ; then mark off from .s along sA the rise of IJ," ; divide 
this into any number of equal parts (say six) in the 
points m, ?i, 0, p, and q. Next, with centre a, radius 
&A, describe a circle, and divide the semicircles ADG and AJG (each 180°) into the same number of equal parts (six) in B, C, D, etc. 
Now, with centre a, radius am, cut «B in h, and with same centre, radius an, describe an arc cutting «C in c, and so on for the points 
d, e, f, g, which are points in the cam to give the upward movement. The next part is to correspond to an interval of rest ; so the 
arc GH of the circle will be that part. As the curve must now draw \" nearer the centre for the next 60°, bisect Aq, and describe 






B^ 


A 

- — '/ 




A. 


<=/ 


y% 




1" 7 
.■> / 


\Ak 


tl 


m 




1'/ V2i-— A_ 


m\ 


D 


M 


/ 






< 


% 


/ 


^-30° \ 


w 




F\ 


.;i^>j 




H 



Fig. 240. — Cam used in shearing machine. 



Pig. 241. — Cam, heart type. 



arcs as shown, to give the points i and j. 



' Obviously the larger the number of divisions the more accurate will the curve be. 
templates, or by the movement of a lath on a cylinder. 



Next, with centre a, radius as, describe an arc cutting aJ in s ; divide sj, the remaining 

Of course, this curve could be easily described mechanically by means of 



SPIRAJLS AND MISCELLANEOUS CURVES 



83 



distaBce the point is to fall througli, iuto three fthe numbers of the remaining divisions of tlie circle) equal parts in v and iv. Then 

describe arcs through these points to give the points k, I. A flowing line through the points a, h, c, d, c, as shown dotted, will give 

the required curve. 

Of course, the reciprocating point would only move in the line SA in the way explained when it had no magnitude. In the 

case that occurs in practice the point is replaced by a roller ; then the curve has to be slightly modified, as shown in contact with 

the small circle above a, whose centre is S, aud whose diameter equals that of the roller, the cross-hatched figure representing the cam. 

NoTB. — As by a skilful arrangement of suitable cams almost any desired movement of a machine part can be arranged, the student would do weU to work 
out a few examples. 

156. To set out a Simple Cam (another Example). — Look at Figs. 242 to 244. You see three positions of a cam in relation 





>-L0 




-2 



•LO 




Fig. 242.— Bottom Fig. 243.- Interme- Fig. 244.— Top 
position of slider, diate position of slider, position of slider. 



Fig. 245. — Bottom position 
of slider. 



Fig. 246. -Intermediate 
position of slider. 



Fig. 24Y. — Top position 
of slider. 



to a slider MN. As the angle da is 60°, it is obvious that as the cam rotates uniformly the slider is rising a cUstance from a to 5 
whilst its roller is in contact from d to h. Three positions of the slider are shown in the Figs. 242 to 244, and these should speak 
for themselves. In setting out the cam, first mark off the angle of 60° and the lift ah on the radius produced through a, divide 
ab and ad into the same number of equal parts (in this case two), and with centre of the circle C, radius C to the point between 
a aud h, describe an arc to cut the radius 01 produced in e. Through d, e, and b draw a flowing line to complete the cam. Of 
course, if the end N of tlie slider MN be fitted with a roller, the edge of the cam must be set back a distance equal to the radius 
of the roller, as in the previous problem. You will notice that with this arrangement the edge of the cam causes an oblique thrust 
to come on the slider, which then tends to bind it in the guides. Let us see how this can be avoided. 

157. Involute Cams. — Look well at the Figs. 245 to 247, and keep before your mind that the object of the arrangement you see 



84 



INDUSTRIAL DRAWING AND GEOMETRY 



12A 



is to convert the rotary motion of the cam into an up-and-down sliding motion of the slider MN". This part is in its lowest 
position in Fig. 245, and in its highest one in Fig. 247, where it is on the point of disengagement. When this occurs, of course the 
slider falls by gravitation to its lowest position again, only to be again raised when the part d comes into position under the 
projecting piece P. The lower surface of this part is horizontal, and for all positions whilst in contact with the cam, this surface 
is tangential to the curve deb, as can be clearly seen in Fig. 246. If you have carefully read Problem 154, you will uaderstand how 
to set out the involute curve dch, and will see that the normal to the surface of the curve at the point of contact is always vertical, 
and tangential to the circle ; so that no matter how great the load, there is no tendency on the part of the cam to put a side 
thrust on the slider, as in the previous example. 

158. The Spiral of Archimedes. — In this spiral consecutive points in the curve uniformly recede from the pole; that is to 

say, the length of the radius vector is directly proportional to 
its position or vectorial augle.^ This will be understood by 
reading the following problem. 

159. To describe One Convolution of the Spiral of Archimedes, 
the Pole, the Initial Line, and the Length of the longest Radius 
Vector being given. — Let P and PA (Fig. 248) be the given pole and 
initial line respectively. Then from P along PA mark off PA, equal to 
the largest radius vector. Then with P as centre, radius PA, describe a 
circle. Next divide the circle and the line PA into any number of equal 
parts (say 12), as shown in the figiu-e. From the centre and pole P di-aw 
lines to the points 1, 2, 3, etc., dividing the circle. Then with centre P, 
radius P 11', describe an arc cutting P 11 in Z, and with the same centre, 
radius P 10', describe an arc cutting P 10 in h, and so on for the other 
points, j, i, h, etc. A flowing line drawn through these points fi'om A 
wUl terminate in P the pole. 

Notes. — 1. It will be seen that the radii vectores. Pi, Vk, Fj, etc., get 
shorter by the fixed distance of ^'^ of PA for each angle of yL of a revolution 
passed through. This being the case, it is obvious that if a cam were 
arranged this shape, it would give a uniform reciprocating motion when 




Fig. 248. — Arohimedian spiral. 



Pig. 249.- 



G B 

Curve of sines. 



The part of the curve Fbcdefg, if reproduced on the other side, making a figure symmetrical about the 

e, etc., along the respective radii vectores, the 



uniformly revolving about an axis through the pole P. 
line Fg, gives the form of a cam called the Heart cam. 

2. The points BODE, etc., are in a second convolution of the curve, and are found by marking off from b, c, d. 
distance PA. The curve can in this way be carried on to any extent, as the convolutions are parallel. 

3. In the centrifugal pump the casing takes the form of this curve for a constant velocity, as the section must be proportional to the quantity passing through 
in a given time. See Goodman's " Mechanics Applied to Engineering," p. 546. 

160. The Curve of Sines, or Harmonic Curve. — This is the curve in which a musical string vibrates. The ordinates in this curve are proportional 
to the sines of the angles, which are the same fractions of 360° as the corresponding abscissEB of the wave-length. 

161. To draw the Curve of Sines, the Amplitude and Length of a Vibration being given. — Let AC (Fig. 249) be the given amplitude and 
EC the length. Then place AC at right angles to EC, as shown in the figxure. With C as centre, radius AC, describe the semicircle AcB. Next divide 
the semicircle into any number (say six) of equal parts in a, 6, c, etc. ; also each half (ED and DC) of the length EC into the same number of parts in 1, 



When the radius vector is inversely proportional to the vectorial angle, the curve generated is called the reciprocal, or hyperbolic spiral. 



SPIRALS AND MISCELLANEOUS CURVES 



85 



2, 3, etc. Tlien through the points 1, 2, 3, etc., draw ordinates to cut abscissa through a, b, c, etc., as shown. A flowing line drawn through the points 
of intersection is the required curve. 

It will be noticed that the radius of oui-vatui-e increases from the crests F and G- to D, the point of inflection, where it becomes infinite. The 
distance ED is half a wave-length, which is repeated from T> to C, the other side of EC. 

Note. — If the point 6 move uniformly in the circle, the point P, its projection on AB, will move in AB with a simple harmonic motion. The satellites 
of Jupiter as seen from the earth nearly have this motion. 

162. The Helix. — If you cut out of a piece of paper a right-angled triangle ABC (Fig. 2.50), making its base AB equal in 
length to the circumference of a cylinder = dir, where d is its diameter, and wrap the triangle round the cylinder, the hypote- 
nuse BC will form a curve BE2C2 (winding round the cylinder) which is called a helix (or erroneously a spiral). On comparing 
Figs. 249 and 251, you will see the curve GDFE of 249 is exactly similar to the curve BG2E2C2 of 251, but the former is a plane 




11 10 9 8 7 6 S + 3 2 I I 

CtrcitMuerence =air n 





Fig. 250. — Development of a helix. 



Fig. 251. — Projections of a helix. 



Fig. 252. — Plotting paper for polar co-ordinates. 



curve, and its axis is in a horizontal position, whilst the latter is the elevation of a line winding round the cylinder, and its axis 
is in a vertical position. 

The plans of the cylinder and helix are the circle in Fig. 251. A2B2, the plan of the triangle, in being wound round the cylinder, 
moves in the direction of the numbers 1, 2, 3, etc., in the circle, and the other features of the construction should now speak for 
themselves. 

163. Screw Threads are Helices. — -Thus, if the helix in Fig. 251 represented an edge of a screw thread, such a one as in 
Fig. 400 for instance, the distance BC2 would be the pitch of the thread or helix. 

164. Plotting Paper for Polar Co-ordinates.^ — Fig. 252 shows to a reduced scale a circle divided by concentric circles and radial 

' This paper can be bought at some instrument shops. Messrs. Bemrose & Sons make a speciality of it, selling it in pads of 40 sheets (diameter of outer 
circle SjI" or 135 mm.) for a shilling, with tables included. 



86 INDUSTRIAL DRAWING AND GEOMETRY 

lines ; the latter divide the circles into angles of 5°, and if the concentric circles be equally spaced, say J" apart, you can readily 
understand how useful such a diagram is in working problems on spiral and such curves as we have been dealing with in this 
chapter. 

Why not try and draw such a diagram, making the great circle 6", say, spacing the circles J" or 3" apart? You could then 
quite easily set out the Archimedian spiral shown in Fig. 248, and some interesting cam forms like those in Figs. 241 to 247. 

EXERCISES. 

Typical Obal Exercises. 

1. If you unwind a thread from a reel of cotton and make the end of the thread guide a pencil on a piece of paper pressed against the end of the reel, what 
is the name of the curve that will be traced ? 

2. Whilst a man is walking at a uniform speed from the centre of a locomotive turntable to its outer edge, along a radius of the circle, the table makes a 
complete revolution. What is the name of the curve the man would be moving in in relation to the ground ? 

3. If you cut out a paper right-angled triangle and wrap it round a cylinder, ruler, or piece of broomstick, starting with the base at right angles to the axis of 
the cylinder, the hypotenuse will form a curve winding round the cylinder. What is the name of this curve ? Can you call to mind any well-known bodies 
or machine details on which the curve is to be found ? 

Deawing Exercises. 

4. Draw the involute of a 2" circle, and at a point in the curve 2" from the centre of the circle draw a tangent. 

5. The largest radius vector of an Archimedian spiral of one convolution is If". Draw the curve, and a tangent to it making an angle of 30° with the 
initial line. 

6. Draw a cam so that a double uniform reciprocating motion through li" in a straight line is given to a point by the cam revolving once. 

7. Set out a cam to give the following motion in a straight line ; For the first 60° motion, uniform rise of 1", interval of rest during next 45°, uniform faU. 
of A" during the next 45°, interval of rest during next 60°, and uniform rise of 1" and fall of IJ" during the following 150°. Make the diameter of the roller J", 
and the part of the cam nearest the centre or axis h" from it. 

8. Set out a curve of sines, making the amplitude li", and the length 3". 



CHAPTEE XVI 

THE APPLICATION OF GEOMETRY TO ORNAMENTAL AND DECORATIVE DESIGN 

165. Introductory Reraarks. — The beautiful artistic creations of the ancient Greeks appear to have been always controlled and perfected by appli- 
cations of tlie laws and priucijiles of geometry, and the decline of art iu different ages can be traced to the neglect of these principles. In fact, it can 
be shown that whenever the geometrical spirit ceased to influence design, the decline of art was rapid. Few workers realize how universal are the 
applications of geometry to artistic design, and how the most beautiful forms, from a simple pattern to a most elaborate design giving the greatest 
charm, can be produced by the artist who has mastered the art of using geometry as an instrument to assist him to devise, arrange, and combine some of 
the simplest geometrical figures. Obviously, the possibilities in this direction are without limit. The ai-tist who confines himself to the manipulation 
of geometrical figures, or to forms that have a geometi'ical formation or foundation, without entering the realms of pure art, gets many of his ideas 
from natural objects, such as shells, leaves, buds, and flowers, which he conventionalizes in such a way that they lend themselves to geometrical treat- 
ment. Without attempting to particularize the great variety of work that comes within his province, the foUowiug may be mentioned as being among 
the most important : Geometrical patterns and simple tracery,' for decorative purposes, such as tessellated mosaic work for floors, floor-cloths, 
parquet, wall-papers, carpets, rugs, marquetry, buhlwork, etc. ; encaustic tiles for walls, floors, etc. ; mural decorations ; tracery of Gothic and other 
windows (which has given such characteristic beauty to the architecture of the fourteenth century) ; painted and sculptured patterns on vases, ivory, 
pottery, and porcelain ; the ornamental treatment of glass and jewellery ; the arches of bridges and ecclesiastical and other buildings ; furniture, iron- 
work, such as gates, railings, grOles, and medieval ironwork, including hinges of doors and church chests, ironwork of windows, etc. ; works in bronze, 
and other metals, etc. 

Geometeical Pattekns. 

166. Marquetry, or Buhl Work (Fr. marquetrie), is inlaid work consisting of thin layers of coloured woods or ivory glued on to a backing of 
oak or fir, well dried and seasoned, which, to prevent warping, is composed of several thicknesses. The art was cultivated by the early Italian cabinet- 
makers, who rejiresented by its means not only geometrical patterns, but landscapes and figures. 

167. Mosaic- — The filling up of a plane surface with small pieces of marble, opaque glass, coloured clays, or other substances, so as to form a 
pattern, was practised by the Greeks in the fourth century B.C., the best Hellenic examples of this kind of work being discovered during the excavations 
at Olympia about 187-5. Some very fine specimens of mosaic are to be seen in the Chapel of the Confessor and in front of the high altar at Westminster. 

When the design is formed of small cubes, generally of marble, it is called tessellated, and when formed of larger pieces of marble or glazed 
earthenware, shaped and cut or formed so as to fit one another accurately, sextile.' 

Note. — In laying down a pavement of mosaic or inlaid work on which persons are to walk, too many summits (or corners) should not meet in any one point, 
as any considerable weight on that point may injure the texture and soliaity of the work. 

' You must give a good deal of attention to the study of some of the preceding chapters before you can hope to draw tracery well. 
^ Greek for " small stone." 

^ When pieces of opaque glass are used to form complicated pictures for the ornamentation of walls and vaults, the design is fictile or vermioulated. 

87 



88 



INDUSTRIAL DRAWING AND GEOMETRY 



We may now proceed to examine a few examples of the application of geometry to ornamental and decorative design. Many of them have been 
selected from the author's work on " Geometrical Drawing," and in not a few cases designed to illustrate a variety of applications which will be referred to 
as we proceed. You should look upon many of them as examples that can be varied in an infinite number of ways, and you should, after drawing them as 
exercises (preferably to a larger scale), endeavour to devise suitable variations, realizing that in this direction there is boundless scope for the exercise of 
your ingenuity and taste ; for, as has been truly said, geometry is the handmaid of ornament. 

Although in nearly each of the following figures an attempt has been made, by leaving a part unfinished and by the use of dotted lines, to show how 
it has been constructed, it is assumed that you have read the preceding chapters, which have a bearing on this part of our work, and you should therefore 
experience no difiiculty in reproducing them. 

168. Some Points relating to the Application of Simple Figures in Mosaic — Paving, Glazing, and generally to all Inlaid Work and 




A/i|l|'!l!i/'iiiri"i|'ii! 



D E F G 

Pig. 254. — Second step. 



B A 




Pig. 255.— Third step. 



Pig. 256. — Complete pattern. 



Geometrical Patterns. — These branches of industry are concerned in covering or filling a given area with figures terminated by 
straight lines. 

If it is a condition that all the figures should be regular, and have the same number of sides, there are only three figures avail- 
able, namely— 

1st. Equilateral triangles, the summits or apices of which (six in number) meet at the same point, as in Fig. 256. 

2nd. Squares, the summits of which meet, four and four, at the same point, as in Fig. 274 (the dotted squares). 

3rd. Hexagons, the summits of which meet, three and three, at the same point, as in Fig. 282. 

But, in addition to these regular figures, there are many more or less regular, which, when combined together, produce a pleasing 
and artistic appearance, as you will see. 

169. Exercise — Simple Patterns formed by Equilateral Triangles. — Some interesting patterns can be drawn with the assistance of 
your 60° set-square and tee-square. Thus, if you prick off with your compasses a few equal divisions (say, i") along a line AB 
(Fig. 253), and with the set-square complete the equilateral triangle ABC. Then through these divisions, D, E, F, G, draw 
lines (Fig. 254) DH, EI, etc., with the 60° set-square. Reversing the square (Fig. 255) draw lines GJ, FK, etc., and (Fig. 256) 
through the apices of the small triangles draw lines OP, QR, etc., parallel to AB, to complete the figure or pattern. Or a rectangle 



THE APPLICATION OF GEOMETRY TO ORNAMENTAL AND DECORATIVE DESIGN 



89 



can be filled with the small triangles (as shown dotted in the figure) if its height = ~-AB = 0-866 AB. 



If the sides of the small 



triangles (or the rhombuses, as the case may be) are produced- till they cut the sides of the rectangle, you may then shade or colour 
some of the triausles to get effect, or vary the pattern according to your taste and ingenuity. 

170. To draw a Diamond Chequered Pattern. — You will experience no trouble now in making such drawings as are shown in 




^ 


ia 


^^ 


-j^ 


^^ 


?f^' 


^^ 




'^^'^'^ 


"^^. 


^¥^ 


^^ 


^ 




^""^^^y 


-i^ 


^^S 


?^^ 


^^ 




^<^^ 


^^^ 


^^ 


^^^ 


^ 


^ 




^^ 


*^ 


#^ 






'"■'^^ 


^^ 


^^ 


^:^' 


1 _i 




<c^_ 


— — ^i& 


a&-^ — _ 


IlS&o 



Fig. 257.— First step. 



Pig. 258.— Second step. 



Pig. 260. — Design for cast-iron Igrating. 



Pig. 259.— Finished pattern 
of chequered plate. 

Figs. 2.57 to 2.59, which should speak for themselves. This particular design of a chequered plate is called the Admiralty pattern,^ 
the rhombuses having acute angles of 60°. 

Cast-iron Grating. —Fig. 260 shows a diamond chequered pattern that is much used for gratings. You will now experience 
no trouble in setting out this pattern. 









Pig. 261. 



Fig. 262. 



Pig. 263. 



Pig. 264. 



Fig. 265. 



Pis. 266. 



171. Simple Star Forms are readily drawn with the 60° set-square, as will be seen from an inspection of Figs. 261 to 265, 

which should now speak for themselves. 

Note. — It will be noticed that the base ci of the triangle (Pig. 262) is divided into sis equal parts, and that a lineaft, the second from the end c, will enable you 
to draw through h the side ef of the second triangle. A circle with an inscribed equUateral triangle is shown in Pig. 266. If three other triangles be placed over 
this, as shown in Pig. 267, with apices at B, D, and E, either of the star forms (Figs. 268 or 269) with 12 points can be produced. 

' As you will see from Fig. 259, the pattern represents a plate with raised strips ; such plates are used for engine and boiler-room flooring and such purposes, 
to prevent the feet from slipping, and they are generally made of mUd steel. 



90 



INDUSTRIAL DRAWING AND GEOMETRY 



The 8-pointed stars (Figs. 271 and 272) are commenced by first drawing the circumscribing circles, and using the 45° set- 
square, as shown in Fig. 270. 

The Rosette (Fig. 273) has the diagonals and cross lines of the circumscribing square for guiding or construction lines. 

172. Patterns suitable for Tiles, Linoleum, etc., based on squares, can be made to have a very pleasing effect, such as Fig. 274, and 
a little ingenuity will enable you to draw a variety of these. A variation in form can be made by arranging the squares with their 




Pig. 268. 



Fig. 269 



Fig. 271. 



Fig. 272. 



Fig. 273. 



diagonals parallel to the sides of the enclosing rectangle, as in the trellis pattern, Fig. 279. Some very pleasing effects can be got 
by drawing suitable patterns on the squares as in Fig. 275. A unit of the pattern is formed by the squares ABCD, EFGH, or 





Fig. 274. 



Fig. 275. 




Fig. 276. 



by those whose corners are at the centres of the stars. Another type of these patterns is shown in Fig. 276 ; it is obviously also 
based on squares as shown. 

173. Greek Frets. — These are based on squares, which should be first drawn, then a variety of interesting fret patterns, such as 
those shown in Figs. 277 and 278, can easily be set out. You should get a sheet of squared paper, and practise drawing some. 



THE APPLICATION OF GEOMETRY TO ORNAMENTAL AND DECORATIVE DESIGN 



91 



174. Trellis Patterns. — The trellis pattern in Fig. 280 you will see, upon examination, is based on squares as arranged in 
Fig. 279. If these are drawn first the figure is easily completed. 



^ ^ ^ 



iS] 



Fig. 277. — Greek fret based on squares. 




Pig. 279. — Arrangement of squares as base for trellis, etc. 



t 


/\/\/^ 


\A/V\ 


/\/ 


\/\ 


to 


\/\/\/\/\/\/\/\/\ / 


00 


A/\A/vVv\/\A 


o 


\ / x/N/VW Y \/\/ 


X 


AAA/\A/\A\AA 


I/) 


\ A/vy^AvAA A / 


ca 


/\A/Vy YY \/\A 


I 


\/V\A/\/\ A A A/ 


1 


/ \a/ 


v/V\A 


A/ 


\/\ 



BIHIBJIBIE 



Fig. 278. — Greek fret, key patterui based on squares. 




Fig. 280.— Trellis pattern. 





Fig. 281.— Base triangles, first step. Fig. 282.— Complete pattern of tiles. Fig. 283.— Pattern for tiles, etc. 

175. Hexagons in a Rectangle. — Figs. 281 and 282 show two steps in drawing a pattern formed by hexagons, and based on 



92 



INDUSTRIAL DRAWING AND GEOMETRY 



equilateral triangles, as in Fig. 256. These should now speak for themselves. Two variations of these are shown in Figs. 283 and 
284. A little schemiug will enable you to devise many others. 

176, Use of Guiding Lines in Detail Work. — You have had your attention called to the decorated squares in the pattern (Fig. 275). 



TV— \ — ^ 





FiQ. 284. — Pattern for tiles, etc. 



A rs 

Pig. 285. — Unit of a pattern 



/^ ^ 


/ \ 


^ 


" ^ 


A. 


A 


\ 


V I 




V 


V 


A 


/ 


J 


K ^ 


\ ^ 


^ 


- ^ 



Pig. 286. — Showing use of construction lines. 



In setting out such forms it is often helpful to make use of such lines as form the five squares in Fig. 285, or the diagonals and 
inscribed square in Fig. 286 for guiding purposes. 






Fig. 287.— Hexagonal tile. Pig. 288. Pig. 289. — Decoration of a circular plaque. 

Further, the hexagonal tiles in Fig. 282 may be decorated in a variety of ways : such as shown in Fig. 287, for example. The 



THE APPLICATION OF GEOxMETRY TO ORNAMENTAL AND DECORATIVE DESIGN 93 

sides in tliis case being trisected, and lines drawn through the points parallel to the sides, shading the triangles, as shown, giving 
a central star. 

Fig. 288 shows an elegant interlacing pattern suitable for the decoration of square or circle ; and Fig. 289 one quarter of the 
decoration of a circular plaque. '^ Of course, all the panels would be decorated as the centre one is. 

EXERCISES. 

Typical Okal Exercises. 

1. Small pieces of marble, opaque glass, and coloured clays are sometimes used for fancy paving purposes. What name is giving to such work ? 

2. What is the difierence betvpeen tessellated and sextile pavements ? 

3. In laying down a pavement or inlaid work on which persons are to walk, what practical objection is there to making many summits or corners meet 
in the same point ? 

i. A certain kind of inlaid work consists of thin layers of coloured wood or ivory glued ou a backing of wood. By what name is this work kno^vn ? 

5. If it is a condition that all the figures forming a geometrical pattern shall be regular and have the same number of sides, how many regular geometrical 
figures are available ? 

Dbawing Exebcises. 

6. Draw an equilateral triangle on a 6" base and cover its surface with 1" equilateral triangles. Count these and compare the areas of the whole figure to 
that of one of the elements. 

7. Draw a rectangle with 5" base and 4-33" height, and cover it with J" equilateral triangles, as in Fig. 256. How many of these triangles, and how many 
half ones are there ? Compare the area of the rectangle to that of one of the triangles, i.e. give the ratio of areas. 

8. Make a drawing of a piece of diamond chequered plate (Fig. 259) 5" X 4", making the size of the equilateral triangles upon which it is based 1". Note. — 
This is known as Admiralty pattern. What is the object of the ridges or raised strips on the surface of the plates ? 

9. Draw the three stars shown in Figs. 263 to 265, making the equilateral triangle, upon which they are based, with 3" sides. 

10. In 3" circles draw the eight-pointed stars shown in Figs. 271 and 272. 

11. Draw a SJ" square and in it set out the tile pattern shown in Fig. 274. 

12. Draw the Greek fret key pattern (Fig. 278), making its breadth 2J" and its length about 8" or 9". 

13. Draw the trellis pattern shown in Fig. 280, making its breadth 2" and length about 6". 

14. Set out the hexagonal pattern shown in Pig. 282, enclose it in a square of about 3" side, and make the triangles, upon which it is based, with J" sides. 

15. Draw a rectangle of 5" base and 4-83" in height, and set out the pattern in Fig. 283,-usLng i" equilateral triangles. 

16. Draw a rectangle of 5" base and 4-33" in height, and set out the pattern in Fig. 284, using J" equilateral triangles. 

17. The unit of a pattern is shown in Pig. 285. Set out the pattern showing at least 9 units, and making each 1^" x IJ". 

' Taken from a Board of Education Examination paper, by bind permission of H.M. Stationery Office. 



CHAPTER XVII 

PLAN AND ELEVATION— HOW TO MAKE A WORKING DRAWING OF A SOLID BODY. 

177. Introduction. — We will assume that you have carefully read at least the first three chapters, particularly Chapter II., and 
that you are about to attempt a drawing of some simple object. Now, before you can do this intelligently, it is obvious that you 
should have a fair acquaintance with elementary projection, and as it is possible that you may not have received instruction in this 
useful branch of geometry, we will proceed to briefly explain how an object may be drawn in plan and elevation ; for the shape and 
proportions of most simple solids can be completely shown by drawing two views only, namely — 

178. Plan and Elevation, called their -projections. The terms " plan " and " elevation," as applied to the representation of an 
object, are fairly well understood in a general way. Thus we speak of the elevation of a house, meaning the view we get by 
looking at its front, back, or sides. ]'>y such a view we see its height and breadth, and the height of everything shown is found 
on this clcvational view. Again, we speak of the plan of a plot of ground. This view, of course, shows its length and 
breadth, and the distance it may be from some landmark. In the same way the plan of a house or any object is the top 
view we get by looking down on it from above. All this and much more can better be made clear by referring to an example ; 
and as first steps cannot be made too easy, the subject frequently presenting considerable difficulties to beginners, you cannot 
do better than take a sheet of drawing paper and any rectangular solid, such as a box or a book, and work out the following 
simple exercise : — 

Let BACD (Fig. 290) be the sheet of paper. Draw across it any line XY (this may be done in the ordinary way with the 
T-square), and place the bottom (EFGK) of your box on the paper, so that one of the long edges, EK, is resting on XY. Then 
bend the part of the paper BD about the line XY, as shown, until it touches the back of the box EKIJ. If, when the paper is in 
this position, a pencil point be drawn round the box, marking the lines EFGKIJE, we shall have on the horizontal plane (XYCA) a 
plan EFGK of the box, and on the vertical plane (XYD'B') an elevation EKIJ. Now let us suppose that we are to draw the plan and 
elevation of the box in its present position, in the ordinary way. Begin by drawing XY (Fig. 291) with the aid of the T-square; 
then construct EKIJ (the elevation), a rectangle, making EK equal to the length of the box, and EJ equal to its thickness, 
remembering that EK must rest on the ground line (XY), as the box is resting on the ground (horizontal plane); and that as it is 
touching the vertical plane, the plan, which may now be projected (carried down) from the elevation, must be drawn showing the 
back EK of the box touching XY. Of course, all the lines on the plan and elevation are drawn with the assistance of the T-square 

94 



PLAN AND ELEVATION— HOW TO MAKE A ^VORKING DRAWING OF A SOLID BODY 



95 



and the set-sqnare S. You will of course notice that in this case the plan might have been dr 
it. That is to say, tliis is a case where either the plan or tlie elevation may be first drawn, 
not a matter of choice.) It will now 



awn first, and the elevation projected from 
(Cases will occur directly where this is 



be 



seen 



that 



Kg. 



291 we have re- 




presented the form and position of a 
body which possesses three dimensions 
(namely, length, breadth, and thickness) 
upon a plane having only two dimen- 
sions, namely, length and breadth. You 
should now bend the paper (Fig. 291) 
about its XY, so that the two parts are 
at right angles, as in Fig. 290, and then 
imagine that the box is in its place, as 
it is shown in that figure ; for beginners 
frequently fail to make much progress 
owing to their inability to exercise their 
imagination in this way. 

As a farther exercise we may draw 
the plan and elevation of a rectangular 
bloclc in such positions as shown in 
Figs. 292 and 293, where it will be seen 
that the two views in the latter figure 
are separated by the distance aa\ and 
to enable you to see what bearing this 
change of position has upon the pre- 
vious case we will proceed to work a 
little problem which shall be a distinct 
step in advance of the previous study, 
but, nevertheless, one that ought to be 
readily understood. The problem may 
be stated thus: — 

179. To draw the Plan and Elevation of a Rectangular Block 9" long, 6" wide, 
horizontal, and 1" above the H.P. (or Ground) and One of its Sides is parallel to the 
half-size.) — First draw across the paper a line, and mark it XY ^ (Fig. 292). Then fold or 
previous study, and as shown in the figure, and place the block on something 1" thick ; it 

' This is the ground-line, as it is called ; it is invariably marked XY in 





J 


! 








E 


K 


N 












\ 


Y 










s 


\, 




F 


G 


° ) 







Fig. 290. — Relation of plan to elevation. 



Pig. 291. — Projecting one view from the other. 

and 3" thick, when a 9" x 6" Face is 
Vertical Plane, and 2" from it. (Scale 
bend the paper about tliis line, as in the 
will then be the right height above the 
geometry. 



96 



INDUSTRIAL DRAWING AND GEOMETRY 



ground, or horizontal plane. If we now move it till its back face is parallel to the vertical plane, and 2" from it, the block will be 
in the required position. The figure clearly shows this position, and at this stage it will be instructive to compare this problem 

with the previous study (assuming that the box and the block are the same 
size). It will be noticed that the plan in Fig. 292 is the same shape as the 
plan in Fig. 291 (this must be so, as both solids are horizontal), but is 2" 
distant from XY (that is, 2" from the V.P.), and similarly with the elevations, 

they are the same shape. The one in Fig, 




a' 



I 



ELEVATION 



I" 



2" 

4- 



9" 
PLAN 



Pig. 292.— Block in position, between folded drawing paper. Fig. 293.- 



292, being 1" above XY, shows that it is 
1" high. Of course it will be noticed that 
the lines (projectors) connecting the block 
with its elevation are perpendicular to the 
V.P., and also the lines connecting the 
block and the plan are perpendicular in 
the horizontal plane. The figure also shows 
by dotted lines the paper folded (con- 
structed) back into its proper (normal) 
position, and the dotted elevation shown 
will be seen to be in the same straight line 
ivith the plan, perpendicular to the ground- 
line (XY). Thus, when the projections of 
an object are drawn, we always have the 
plaa and elevation in the same straight line 
perpendicular to the ground-line, 
knowing exactly how the views will appear in shape and 

The first thing to do is to draw 
Remembering 



6 



-Projections of a rectangular 
block or cuboid. 



To make this second study complete, let ns suppose that we 
position, wish to draw in the ordinary way the projections of the block to satisfy the problem. 
XY, the ground-line (Fig. 293). Then, as in this case we can first draw either projection, let us start on the plan, 
that the block is 2" from the V.P., we draw a line ah parallel to XY and 2" below it, and on this line we construct the plan, which 
of course is a rectangle, -whose length is 9" and breadth 6". Then from each end of this plan draw a projector perpendicular to XY ; 
between these projectors draw a'b' the bottom of the elevation, parallel to XY and 1" above it, and on this line complete the 
rectangle, whose breadth is 3" (the block's thickness), which forms the elevation. The projectors are best drawn undotted, but much 
thinner than the lines that form the projections.^ 

This completes the projections, and you would do well to repeat the operation explained in the previous study, and try to 
imagine that the solid itself is standing over the plan, and in front of the elevation, as shown in the figure. 

Note.- — Before leaving this study, we might notice that the line a'b' on the elevation represents the bottom of the block, a horizontal surface, a surface 

' In an ordinary mechanical drawing the projectors are not allowed to remain, any that may have been drawn as a matter of necessity or convenience being 
rubbed out. 



PLAN AND ELEVATION— HOW TO MAKE A WORKING DRAWING OF A SOLID BODY 



97 



SECTION ON 
LINE CD. 



10 
ELEVATION ^ 



END 
ELEVATION 



tu . 



ELEVATION 



3 



-3 
Fig. 294, 



PliAN 



■B 




-Projecting sections and end 
elevations. 



Fig. 295.— Projections of a 
triangular prism. 



perpendicular to the vertical plane. You wUl directly better understand that the projections on a plane of all surfaces perpendicular to it are straight lines on 
that plane. Thus the line ah on the horizontal plane is the plan of a vertical side. 

180. End (or Side) Elevations, and Sections. — Let us suppose we are looking at the rectangular block (Fig. 294) in the direction 
of the arrows A and B ; the view we then get is called an end elevation, and it may be shown as at E, where the figure is obviously 
constructed with the assistance of the plan, the 3" height being 
marked off witli the dividers. It is generally more convenient to 
place this view by the side of the elevation, as shown at AF ; the 
view is then projected from the elevation as shown, the 6" breadth 
being ma'i-ked off with the dividers or found by using the arcs/m 
and hii. If we were to cut through the solid with a vertical saw- 
cut along the line CD in plan, the true shape of the cut would be 
a vertical section (a section on the line CD as it is called) of the 
solid. This is shown at G in the position which is usually most 
convenient in relation to the elevation. It is drawn in the same 
way as the end elevation AF. 

Usually an end elevation becomes necessary because some part or parts cannot be properly seen either in plan or elevation ; or, 
taking the simplest case, it may be because one or more of the edges of the solid is not parallel to the H.P. or V.P., and therefore will 
not be seen in true length either in plan or elevation. You will better understand this if you look at Fig. 295. The end elevation, 
read with the other views, shows that the solid is a triangular prism, although its plan and elevation are the same shape as those of 
the rectangular block in Fig. 294. 

We may now proceed to review the salient points as they would probably present themselves to you if you were about to 
commence a working drawing. You should begin by making up your mind as to how many views of the object you intend to show, 
bearing in mind that the drawings should clearly represent the object in such a way that its true dimensions and the form of every 
detail are shown. So long as this is satisfactorily accomplished, as few views as possible should be drawn. Two views at least are 
always required, and these may be an Elevcdion (which shows length and height), and a Plan (which shows length and breadth). 

Or the front elevation and an end elevation may be used to obtain a similar result. But three views, namely a Front elevation,. 
an End elevation, and a Plan are generally shown, with sufficient sectional elevations and sectional j^lccns (part section and part 
elevation, and part section and part plan respectively) to make the external and internal form or construction of the object quite 
clear. The use of dotted lines, as in the end elevation at MM2 KK2 (Fig. 297), for indicating the position of unseen parts, should 
as a rule be avoided as far as possible ; but a judicious use of a few of them may save the making of another view, provided 
always that they do not impair the clearness of the view upon which they are placed. 

Dotted lines should not be used for unseen parts in highly finished coloured drawings, but only for working drawings. In cases 
where the object to be shown is symmetrical about the centre line, it is usual to show one half of the view in elevation, and the other 
half in section, as in the sectional elevation of the coupling, Fig. 476, Art. 248. 

The section may extend slightly beyond the centre line, or may finish at it ; in either case a black line is used to terminate the 
section. This saves the making of a separate sectional view. 



98 



INDUSTRIAL DRAWING AND GEOMETRY 



Although it is obviously desirable to limit the number of views of an object, as previously explained, care must be taken 
not to carry this too far; as in the case of a complicated object, say a casting, much time is often spent by the pattern-maker 
and others in trying to read a drawing, where an additional view or section would have enabled the trained eye to see at a glance 
a mental picture of the required object. 

It is usual in English practice to arrange elevations above plans, or sectional plans, when coavenieat. In American practice this 
is reversed, as we shall see in Chapter XtX. ; but in all cases the views must be arranged so that the relation between two 
adjoining ones may be readily recognized, and so as to facilitate their being properly projected one from another. 

Having' decided upon tlie number of views to be shown, it is usual to take a spare piece of paper, and to rougUy sketch upon it the views decided 
upon in their relative positions one to another, and to mark upon each the overall sizes, as in Fig. 297. The scale to whieb. the views can be drawn, 
in dealing with large bodies, will depend upon the size of the sheet of drawing paper to be used. 

All sheets of drawings forming one set should have equal outer margins, and as far as possible equal spaces between the views. 

Having arranged the positions of the views upon the sheet, and the scale to which they are to be drawn, the next thing to be done is to draw the 
centre lines of the various views. The positions of these can be readily ascertained from a rough sketch used to adjust the spaoings, and they should 
be carefully marked out ; and after this has been done, the various views may be commenced. Of course these remarks are for the guidance of the 
young draughtsman. The beginner will always have plenty of paper to practise on, and need not trouble about the spacing out. 

It is inipossible to lay down any fixed rule as to what view should be first completed; in fact, it is usually the practice 
to work upon two or three views at the same time, drawing some part upon all views first, and then adding another part to these, 
and so on. But generally any known portion, such as the size of a shaft, stroke of a part, leading centres or outline is first 
drawn; and alivays the view from which the greatest number of parts of other views can be projected, or the greatest amount 
of information obtained (frequently a section) is then proceeded with. An axiom being to put in outside sizes of work indefinitely 
first, and to fill in all smaller details, as bolts, rivets, studs, nuts, keys, cotters, etc., afterwards. In the case where a part has 
a circular form the circles should be drawn first, and the other views projected from them, and when a number of similar parts, 
as rivets, bolts, and nuts, occur, it is best to put in the small circles of the entire number first, with one setting of the compasses, 
and then the similar lines of each. This will take less time than if each one is completed singly, and en- 
sures a more uniform result. 

It is not usual to show upon working drawings, bolts, nuts, pins, rivets, studs, keys, cotters, rods, 
shafts, spindles, springs and levers in section when a section plane passes through their axes. The reason 
being that it is less trouble to show them in plan or elevation than in section, and it renders the drawing 
more clear. But all these matters can now be more conveniently dealt with as we proceed to explain how 
drawings of a few simple objects may be made, starting with a very easy example and selecting others so 
that they may gradually present to the student further features and expedients in a progressive way. 

181. Drawings of a Cast-iron Bench Block. — The sketch, Fig. 296, shows the form often given to a 

bench block or anvil, such as is often used in an engineer's fitting shop. Cast Iron is used for the block in 

Fig. 296.— Isometric view of preference to Wrought Iron, as it is much cheaper in first cost, and, being harder, is not so easily injured by 

a enc oc". ^ blow. The flat surfaces may be planed, but in some cases the top only is machined, and in others it is 

used rough as cast. In this and the following exercises, the views and scale selected are so arranged as to enable the object to be 

drawn upon a half imperial sheet of paper, viz. 22" x 15". 




PLAN AND ELEVATION— HOW TO MAKE A WORKING DRAWING OF A SOLID BODY 



99 



ELEVATION. 



END ELEVATION. 




SECTION ONLINE tl.O. 



As a drawing example, the four views of the block shown in Fig. 297, viz. a front elevation, a plan, an end elevation, and 
a section on the line no (see also Fig. 296) taken transversely through the centre of the hole and looking to the right (the left- 
hand portion being removed), are to be drawn full size. 

So commence by placing a sheet of paper on the drawing board and pin it down 
taut and flat, as explained in Art. 21. This being a beginner's exercise, we need not 
trouble very much about spacing out the views of the block we -wish to draw, as pre- 
viously explained. If you have followed the previous exercises you will by this time 
be fairly able to manipulate your instruments correctly, and by the exercise of a little 
intelligence will easily draw the plan and elevation of the block ; so, bearing in mind 
the hints previously given as to wliich view to draw first, it will be seen that this is 
a case where the plan should be first set out. Then start by drawing the centre lines 
jk and cd, intersecting in y (Fig. 297), in suitable positions. The length of the block 
should be first set out by pricking off yj and yk with a 4" opening of the dividers, the 
scale being full size. The T-square is then drawn down to about 3j" below jX-, and 
the 60^ set-square is placed upon it and brought into position so that the pencil will 
be in line h. The Kne is then lightly drawn downward, nearly ^ to the T-square ; and 
the set-square is then slid along the T-square, and a line drawn through j in a similar 
manner. Next prick off with the dividers c and d, .3" on each side of y. The T-square 
is then raised to the lower mark d, and the finished line DF is drawn carefully, once 
and for all, between the two vertical lines previously drawn. The T-square is then raised to the upper mark c and a similar finished 
line, CE, drawn through it. Then rub out the extra portions of the lines at CD and EF and pencil in, completing the rectangle 
CEFD. Next draw the vertical centre line Im of the hole in the block, which will be 2^" from the centre of the block ; and take 
the dividers and set them carefully to 5", and prick off points in the sides of the square from its centre, and pencil in the sides J KML 
of the square in the same way as you did CEFD. 

The elevation may now be proceeded with by first drawing an indefinite line PQ, a suitable distance from CE, and a similar 
line NO at the top, 4^' from it. The side lines PN and OQ may now be projected from the plan and drawn their finished thickness. 
The arched opening ET may now be drawn ; first mark up centre line 1m, the height (1") of the arch above the bottom of the block, 
and set the pencil compasses to an opening of 2^" (the radius of the arch), and describe the arc ET as shown. Then from J and K 
in plan project the vertical finished dotted lines as shown, from bottom to top of the elevation, to indicate the position of the hole. 

To commence the end elevation project two indefinite lines UV and WX from the top and bottom of the elevation respectively, 
and draw the centre line e/ in a suitable position. Then mark off 3" each side of this line and draw the finished sides UW and VX, 
completing the outline as before. To indicate the position of the square hole on this view set off eM and «K, ^" each side of e, and 
draw the dotted Unes MM2 and KK2. The dotted lines S1S2 and L2L3 are projected from the elevation, and indicate the position of top 
of the arch part and the intersection of the arch with the side of the square hole respectively. The section on line no is drawn in a 

' As we do not know exactly where to stop, so we always draw it lightly and too long, and rub out what we do not require after its desired length has been 
obtained. This is much better than to draw a line too short, and to join a piece on to make it of the required length, as the joint always shows. 



Fig. 297.— Four views of a cast-iron bench block. 



100 



INDUSTRIAL DRAWING AND GEOMETRY 



similar way aboi;t a centre line yh, the bottom GH being projected preferably from DF of the plan, and tlie sides YG and ZH from 

UW and VX respectively. Of course the height GY is 4J", the same as that of the elevations. As we are looking at the section 

from the left, we shall see the right-hand side of the section. 

The parts actually cut through by the section plane should be section-lined as shown, and as described in Art. 23. And the 

section lines on both right- and left-hand side of the hole should be drawn sloping in one direction only, as it is one piece of metal. 
The section lines used to indicate cast iron are continuous ones (Fig. 524) ; as shown, they are drawn with the 45° set-square, 

restincT upon the T-square. The distance between them, or pitch of the lines, is a matter of taste, and should vary with the size of 

the part to be sectioned ; in this case lines j'f5th of an inch apart may be used. 
They can be drawn by judging the distances by the eye after a little practice, or 
a line can be drawn at right angles to the slope of the section lines, across the 
figure to be sectioned, and equal spaces set off upon it by ticking them off from 
a scale of equal parts, or by using a pair of dividers. To finish the drawing, 
carefully clean off any matter or lines not required, but the centre lines should 
be left, projecting about \" beyond the boundary of the view they -are shown 
upon. The dimensions need not at present be shown on the drawing. The title 
of the drawing should be neatly written (printed) by hand, at the top of the 
Pig. 298.— Illustrating the treatment of sections. drawing, making it clear and brief. 

If you have any difficulty in realizing what the section on line no (or any other section) shows, you are strongly recommended to make a kind of 
pictorial or isometric stetch (refer to Chapter XX.) of the object, somewhat like that shown at " A " in Fig. 298, or better, if you will take the 
trouble to cut a model of the object out in yellow soap, or mould it in putty or modelling clay ; it need not be to scale, but should be roughly proportionate 
in size ; this model you can cut in the desired position to enable you to realize what shape the section would be. If you use a sketch, and have 
difflcvdty in deciding how the part cut by the section plane will appear, place the section Kne upon youi- sketch in the desired position as no ; then rub 
out the forward portion (that to be removed) up to the section line, as shown at " B," and then try to complete the sketch " B," obtaining the data 
necessary to enable you to do so from the other views of the object. Tor instance, knowing the block to be rectangular with parallel sides, you can add 
to " B " the lines YG and HG, Fig. " C." Then you know from the elevations that the hole g-oes rig-ht through parallel to the sides, so you can 
di-aw the lines KK, and MMj, indicating the cut hole. Of course this is only a sketch, but you should have no difficulty in identifying it 
with the section on line no, as given in Fig. 297. 





PLAN AND ELEVATION— HOW TO MAKE A WORKING DRAWING OF A SOLID BODY 



101 



EXEECISES. 

Typical Obal Exebcises. 

1. What is the name you give to the view of the top of a house, as seen from a balloon, say? 

2. Yon look at the windows of a house. What name would you give to the view you get '? 

3. What is the object of drawing an end elevation or side view of an object ? 

4. Why is it sometimes necessary to draw a sectional view of a body ? • 




Fig. 299. 



Pig. 300. Pig. 301. Fig. 302. Fig. 303. 

Pigs. 299 to 30i, illustrating drawing exercises numbers 5 to 10. 



Pig. 304. 



Drawtng Exercises. 

5. In Fig. 299 a pictorial view of a solid is given. Draw its plan, elevation, and end elevation. Full size. 

6. Draw plan and elevation of the solid shown in Fig. 300. 

7. Draw plan, elevation, and a cross or transverse section through the cotter hole of the solid shown in Pig. 301. 

8. A dimensioned sketch of a corrugated iron shed is given in Fig. 302. Draw three outline views of it. Scale J" to the foot. 

9. Draw a plan and elevation of the solid shown in Fig. 303. Is this a case where a third view could with advantage be drawn ? If so, make the'one that in 
your opinion gives the most information. 

10. The pictorial view of a mounted oilstone is shown in Fig. 804. Draw two views of it. Scale J size. 



CHAPTER XVIII 

PROJECTIONS AND SECTIONS OF SOME TYPICAL SOLIDS 



182. We will now proceed to further study the science of projection, so far as solids are concerned, by working a few typical 

exercises that will bring out some of the operations in meclianical drawing which most frequently occur ; and it will be convenient to 

present these in the form of the following problems : — 

183. To draw the Plan and Elevation of a Rectangular Block 3" long, 2" wide, and 1" thick, placing it with its Base inclined 30° to 

the Horizontal, and one of its Sides parallel to the Vertical Plane and 1" from it. — After studying Art. 178, you will experience little 

trouble in working this problem. First draw across the paper the XY (Fig, 305), and bend the 
paper about this line in the way explained (this should be done with each problem until you can 
dispense with such help). N"ow place the block in the required position, and consider which view 
must be drawn first, plan or elevation. It will be noticed that as the sides are parallel to the 
vertical plane, the elevation will be the true shape of the side, and as the top and bottom are 
inclined to the ground (horizontal), the plan will give a foreshortened view of the top and the upper 
end, so that the elevation must be first drawn. To do this, at any point a' on the XY, draw a line a'd' 
inclined 30° to XY, and above it (this may be drawn with the set-square) construct the rectangular 
side, a'b'c'd', of the block, which is the complete elevation. To construct the plan, draw (with the 
set-square) from the elevation the projectors h'i, da, c'c, and d'd, and then draw the line h^d^, 1" 
below XY, and parallel to it ; this is the plan of one side : next draw hd parallel to h^di, and 2" 
(the width of the block) from it, for the other : pencil in c^c, the plan of the top edge, of course this 
will be seen in plan, while aga is the plan of the bottom edge, which is unseen, and is therefore 
shown dotted. Each corner of the elevation is, of course, the elevation of one of the horizontal 
edges. 











^ 


\k 




h' 






> 


\ 


ci' 




\ay 


^r 






Y 




























4 


\ 




a. 






Ca 


h 






PLAN 


1 


2" 


i 


c 


I 




i 





Fig. 305. — Eectangular block. 



Note. — The line h'h^ is a mere projector, and as such should be drawn very fine or dotted, but the other part, 6,6, of 
the line is a part of the plan, and should be a good bold well-defined line ; the same remarks apply to the other similar 
lines in the figure. 



184. To draw the Plan and Elevation of a Hexagonal Pyramid (Axis 3", edge of base 1^"), when its Base is on the H.P., with an Edge 
of the Base inclined 35° to the V.P. — Commence with the plan (Fig. 306), as the base will be seen in true shape. This will be a 

102 



PROJECTIONS AND SECTIONS OF SOME TYPICAL SOLIDS 



103 



hexagon with a side inclined 35° to the XY/ and the opposite corners joined give the plans of the slant edges of the solid. The 

line on the drawing inclined 35' to V.P. should be laid down with the protractor, and the 1^" edge marked on it. The hexagon 

can be constructed on this, and the plan completed. The elevations of the 

six corners of the base must be on the ground-line, as the base is on the 

H.P. The axis of the pyramid, 3", must be set up from the point where 

the projector from the centre of the hexagon meets the ground-line. The 

whole elevation is completed by joining the elevation of the apex to the 

six points determined on the ground-line for the elevations of the six 

corners of the base, as shown. Obviously, two of the slant edges will be 

unseen in elevation, and will therefore be shown dotted. 

185. To draw the Projections of a Tetrahedron, whose Edges are 2^" long, 
when one of its Slant Edges is parallel to the V.P., and a Face is on the H.P. 
— The tetrahedron is to stand with one of its four faces on the ground (Fig. 
307), so the plan will be an equilateral triangle, with its three corners 
joined to its centre, and as a slant edge is to be parallel to the V.P., you 
should, in starting on the plan, construct the triangle with one of its edges 
perpendicular to the XY, as shown. Construct the elevation by drawing 
fi'om the corners in the plan projectors, giving c', h', and d' on the ground- 
line, as the elevations of the corners of the base. The elevation of the 
apex will be somewhere along the projector through a; its exact position 
is found by describing an arc about c' as centre, with 2i" (true length of 
edges) radius, cutting the projector aaJ in a', which is the required point. 
Complete the elevation by joining a'c' and a'h'. 

It should be noticed that the edge AC is parallel to the V.P. (as required), and therefore the elevation a'c' is the true length of the 
edge. Cases occur when neither slant edge is parallel to the V.P. ; then the height of the tetrahedron must be determined before the 
elevation can be drawn. This height can always be found in the way just described by using an auxiliary plane parallel to a slant 
edge, or by a separate construction. If determined by a separate construction, it should be remembered that the height is found 
when a right-angled triangle, having the plan of a slant edge as its base, the true length of the edge as its hypotenuse, and the 
required height (or axis) as its perpendicular, is drawn (this triangle can always be readily drawn, as two sides and an angle are 
known). The triangle is often conveniently drawn on the plan, as shown at 6aA, where «A is perpendicular to ah, and ha is equal 
to he, the true length of an edge. 

Note. — You should carefully examine a wire model of this solid ; beginners often blunder by assuming that the height is eqnal to the length of the edges, 

186. A Cylinder with its Base on the H.P. is cut by a Plane passing through the Top Left-hand Corner of its Elevation, and making 
an Angle of 60' with its Axis. To draw Plan and True Shape of Section (Axis 2V', Diameter 2"). — Draw the projections of the 

' The inclination of a line to a plane is the inclination of the line to its projection on that plane. 




Fig. 306. — Hexagonal section. 




164 



INDUSTRIAL DRAWING AND GEOMETRY 



cylinder as shown (Fig. 308), and through the top left-hand corner of the elevation set off a line (representing the section plane), 
inclining it 60^ to the axis (or sides). As the plans of all sections of a vertical cylinder are circles, and coincide with plan of the 

cylinder, we have only to cross-hatch the plan with dotted lines,^ as shown, to represent the 
plan of the section. The true shape of the section may be found as shown in the figure. This 
projection of the section is drawn by first dividing the plan into any niimber of parts, say twelve 
(which is the most convenient numlDer). This is easily done by manipulating the 60° set-square, 
or in the following way. Divide the circle into four parts by lines parallel and perpendicular 
to XY, and then with an opening of the compasses equal to the radius, mark off a point each 
side of each of these four points. The circle is then divided into twelve equal parts, a, h, c, etc. : 
from each of these points draw a projector to the top of the elevation. It will be noticed that 
with two exceptions one projector is only required ibr two points, as the points h, c, d, e, f come 
directly in front of the points I, k,j, i, h. To proceed with the projections, draw a line perpen- 
dicular to the cutting plane from each point made by a projector cutting this plane, and along 
this line mark off a point, making it the same distance from the section plane as the plan of the 
projection is from the XY. Thus the projector from g cuts the section plane in g', and the line 
g'G is drawn perpendicular to the section plane, and the distance g'G is marked off equal to the 
distance of g from XY. Passing to the next projector, we have/%' the common elevation of F 
and U, and therefore the two distances /'F and A'H on the same line must be made equal to the 
distance between / and XY, and h and XY respectively. In a similar way the points I, J, K, 
L, A, B, C, D, and E are found. Draw a fair line through these points, and the figure (an 
ellipse) will be the required true shape of the section. 
187. Conic Sections. — There are 




Pig. 308. — Section giving ellipse. 




Fig. 309.— Section 
giving triangle. 



Pia. 310.— Section 
giving circle. 



five different figures or sections of a cone (called conic 
sections) due to cutting planes, according to their different positions. The section will be a 
triangle if the cutting plane pass through the apex of the cone and any part of its base, as in 
Fig. 309. And if the cone be cut into two parts by a plane parallel to its base the section will 
be a circle, as in Fig. 310. The other thi-ee sections are the hyperbola, the ellipse, and the 
parabola, dealt with in Chapters XII. to XIV. ; but we will now draw these curves as actual 
sections of a cone. 

188. Given a Cone (Diameter of Base 2.V', Axis 3") with its Base on the H.P., to draw the 
True Shape of a Section made by a Cutting Plane. — First Case : By a Plane parallel to its Axis 
and ?," from it. — Commence by drawing the plan and elevation of the cone, as shown in Fig. 
311 ; then draw AB ^" from C, the centre of the plan, and parallel to XY, cutting the circular 
base in wi and n. The elevations m' , n' of these points can be at once drawn ; they must be on 
XY, as the base is resting on the ground, and they are points in the required section. Through 



C draw a perpendicular to AB, cutting it in d. This point will obviously be the plan of the highest point in the section ; and to 

' The cross-hatching is shown dotted in plan, as the cut surface is covered by the top part of pyramid. 



PROJECTIONS AND SECTIONS OF SOME TYPICAL SOLIDS 



105 



find its elevation, with centre C, radius Cd, describe an arc cutting OP in e ; then through e draw a projector, cutting the side of the 
elevation in e'. Through c draw e'f, parallel to XY and cutting the axis in d', the required point. Next, with centre 0, and any 
radii Cg and CA (less than Cm), describe 
semicircles, cutting AB in J and K. 
These semicircles represent the plans of 
horizontal sections, whose elevations can 
be found by drawing projectors through 
g and h to cut the sides of the elevation 
of the cone g' and h', draw lines through 
these points parallel to XY; then J' and 
K', elevations of J and K on these lines, 
will be points in the required section, as 
they are on the cone, and must be con- 
tained by the cutting plane. Similar 
points can be found on the other side of 
the elevation, as shown in the iigure, or 
by symmetry; and a fair line drawn 
through these points gives the conic sec- 
tion. This section is a hyperbola, for, as 
explained in Chapter XIV., any cutting 
plane parallel to two generators of the 
cone cuts the solid in a hyperbolic section. 

189. Second Case : By a Plane inclined „...,, ... 

to the Axis at a given Angle (say 50-) and ^"'- SH-Section givmg hyperbola. Fig. 312.-Sectioii giving ellipse. 

passing through a Point in the Base. — Commence by drawing the projections of the cone (Fig. 312), and through a bottom corner 
of the elevation draw the section plane, making QO'' — 50^ = 40° with XY, cutting the opposite side in/". Divide this line into any 
even number of parts (say six), and through these divisions, k'a', j'b', etc., draw projectors ; also through these points draw lines 
across the elevation parallel to XY, cutting the sides in 1 1, 2 2, etc. These lines represent the elevations of circular sections of 
the cone whose circular plans can be projected, cutting projectors through a'k', b'f, etc., as shown in the figure. We may now think 
of the lines ak, bj (the intercepts), etc., as the plans of lines contained by the sec ion plane. It will then be obvious that the points 
a, h, c, etc, are the plans of points both on the plane and on the cone; that is to say, plans of points in the required section of the 
cone. If these points be joined by a flowing line, the figure formed will be the plan of the required section. But we require its true 
shape. To get this, we may construct or revolve the figure into the H.P., as shown in the figure, which should by now speak for itself. 

Note. — Of course, both the plan and true shape are ellipses. It will occur to you that these figures could have been drawn direct from their major and minor 
axes, as explained in Chapter XII. And you might try to draw them in this way, but it will require a little care on your part to find the true length of the 
minor axis. 




106 



INDUSTRIAL DRAWING AND GEOMETRY 



190. Third Case : By a Plane parallel to its Side and I" from it. — Commence by drawing the projections, as before (Fig. 313). A 
line parallel to the side n'\' and I" from it will represent the plane, cutting the cone from ?«,'«' to </' in elevation ; and by dividing 

this line into any number of parts (say six) by lines 
?■----_ parallel to XY, the points a, h, c, etc., in the plan of 

the section can be found, as in the previous case, and 
by constructing the section into the H.P. the true 
shape can be found. This, of course, will be a para- 
bola, as explained in Chapter XIII. 

Note. — The dotted figure showing the true shape is arrived at 
by the alternative method explained in Prob. 186, and, as will be 
seen, by using a fewer number of lines ; but the author invariably 
finds that beginners get a better grasp of the geometry of the 
problems by working them as explained, to begin with. 

Of course the distances A,Mj, B|L|, etc., on the dotted figure 
from n'V, the side of the cone, are made eq^ual to the correspond- 
ing points in the plan from XY. 

191. Preliminary Projections. — To construct the 
required plan and elevation, it is sometimes necessary 
to draw preliminary projections. For example, sup- 
pose our problem is to draw the plan and elevation of a 
pyramid when resting on one of its faces. This can be 
done, as in Fig. 314, by first drawing the plan and 
elevation ahcd and a'b'c'd' of the pyramid when resting 
on its base ; and then swinging the elevation over 
until the face ADC (represented on the elevation by 
the line a'c) touches XY. As the pyramid is hinged 
on the edge cd, this line will be part of the new plan. 
The remaining corneis, hi, aj, of this plan are found 
by dropping down projectors from a\ and b'l to inter- 
sect lines from b and «, parallel to XY, in ai and bi. 
Join these points, as shown, and the elevation and 
plan are completed. 
KiG. 3i5.-Use of preliminary plan. Nothing more need be said about the construction 

of this figure, as a glance at it will explain the whole 
operation. So you may now compare it with the alternative method shown in Fig. 315. Here you will see that, instead of con- 
structing (swinging) the face into the horizontal plane (XY), we have constructed the horizontal plane on to the .face (in geometry 
this is, of course, easily done, as the H.P. is represented by a line), and found our new plan by running down projectors perpendicular 




Fig. 313. — Section giving parabola. 



PROJECTIONS AND SECTIONS OF SOME TYPICAL SOLIDS 



107 



to om" new ground-line X2Y.2 ; and by marking off fli the same distance from X2Y2 as a is from XY, and doing the same with the 
other points — in fact, by making the new plan of each point the same distance from the new ground-line as the old plan is from the 
old ground-line — we have only to join these points to complete the plan. 

Notes. — 1. By comparing the two methods, you wiU notice that in Pig. 315 there are four views of the pyramids against three iu 316 ; and when you remember 
that in some cases the object projected is more complicated, you will, no doubt, conclude that the latter method, namely, moving the ground-line instead of the 
solid, is the best one to adopt. 

2. If this little study is properly understood— and it should not be passed till it is — the following problem will be easily worked. 

192. To draw Plan and Elevation of a Cylindrical Roller, when a Face is inclined 55^ to the Vertical Plane (size 3", diameter 
1" thick). — Fu-st draw plan and elevation of the roller, with a face parallel to the V.P., as shown in Fig. 316. Then draw a new 
(auxiliary) gi-ound-line X2Y2, making 55° with the .. 
i'ace in plan, and divide the elevation into a number 
of equal parts (Prob. 186), a, V, c', d', etc!, and find 
their plans, a, b, c, d, etc. Then from these plans 
project the new elevations a", h", c", d", etc., making 
their heights the same above X2Y2 as they are above 
XY. Draw through these points a", b", c", etc., a fair 
line, and the figure is the new elevation of the front 
face. In the same way project the elevation of the 
back face, and complete the projection by joining the 
top and bottom points, and dotting the part of the 
back that is unseen, as shown. 

Note. — The figure a"b"c"d", etc. (part of the new elevation), 
is an ellipse, and is perfectly symmetrical about centre lines 
parallel and perpendicular to X,Y,. Of course, when a circle is 
inclined to the plane of projection, its projection is always an 
ellipse. 

193. To draw the Plan and Elevation of a Cone 
when lying on its Side. — First draw the plan and ele- 
vation of the cone when standing on its base (Fig. 
317). Then draw the new ground-line X2Y2, through 




Fig. 316 



-Use of preliminary or auxiliary 
elevation. 



Fig. 317. — Cone lying on its side. 



the side a'c, and project the plan of the circular base (as in the previous 
problem), which will be an ellipse, and having found a, the plan of the apex, draw from it tangents to the ellipse, and the plan is 
complete. 

194. A Horizontal Line parallel to the Vertical Plane is given by its Plan and Elevation, to determine the Distance from the XY. — 
Let ab, a'b' (Fig. 318) be the given plan and elevation of the line. Draw a second ground-line, X2Y2, at right angles to XY, and from 
a and b run up projectors perpendicular to it, and mark off a"b" along these projectors, making their height above X2Y2 the same 
as their height above the old ground-line XY. The dotted lines in the figure show one way of marking off the height of the new 



108 



INDUSTRIAL DRAWING AND GEOMETRY 



elevation. Clearly the slant distance between the points XY and aJ'V is the required distance, as these two points are the end 
elevations of XY and the given line. 

Note. — This expedient o£ assuming a new plane perpendicular to the V.P. so that an end view of the planes is obtained is most useful when the distance 
between the XY and a plane parallel to it is required. 

X 



ft 


t 


?>' 


\ 


3] 




A 




> 

3) 






, 


V 


X \ 


V. p. 








3J 

o 

z 


a 




7 




AUX 





Fig. 318. — New elevation, showing distance of line AB from XY. 



DEAWING EXEECISES. 

In selecting problems from the following it should be realized that this course in industrial drawing will probably in most cases extend over a period of some 
two or three years. The more difficult exercises are near the end of them, and a good deal of experience in working problems in the projection of solids is 
necessary before these can be satisfactorily tackled. Models of the solids should be freely used when available. 

1. In Figs. 319 to 325 you have the projections of various solids shown, drawn to a scale of J full size. Write particulars of the shape, name of solid, 
dimensions, and position of each one. As an example of what is required, the following particulars relating to Fig. 319 may be considered typical. Name of solid. 
Square prism. Dimensions. Edge of base IjJ" ; length of solid 2|J". Position. A long edge on the H.P. and perpendicular to V.P. ; a side inclined 30° to H.P. 

2. The length, breadth, and thickness of a rectangular solid are 3", IJ", and 1" respectively. Place it in any position in relation to the horizontal and vertical 
planes, draw its plan and elevation, and write particulars of its position in relation to those planes. 

Note. — This is a type of problem of great educational value. The teacher will be easily able to vary it. 

3. The axis of a square prism is inclined 30° to the horizontal plane, and one of its sides is in contact with the vertical plane. Draw projections of the solid, 
axis 3", and edge of base 2". 

4. Draw plan and elevation of a brick 9" x 4J" x 3", when it is resting on an end, and a 9 x 4J surface is inclined 30° to the vertical plane. Scale, quarter 
fuU size. 

5. Draw plan and elevation of a square prism (height 3", edge of base IJ"), when a long edge is on the H.P., and a face is inclined 60°.^ 

' It should be remembered that when the name of neither plane is mentioned, inclined means inclined to the horizontal plane. 



PROJECTIONS AND SECTIONS OF SOME TYPICAL SOLIDS 



109 







Fig. 319. 



Fig. 320. 




J 




Fig. 321. Fig. 322. Fig. 323. 

Note. — These solids are drawn to a scale of quarter full size. 



Fig. 324. 




Pig. 325. 



6. Draw the projeotions of a square prism (height 3", edge of base IV'), when a diagonal of a face is horizontal, and it is resting on one of its short 
edges. 

Note. — You may draw the elevation first, and then place under it the XY, unless you can see a way of working it direct. 

7. Draw plan and elevation of a cube of 2" edge, when a diagonal of a face is inclined 80°. 

8. Draw projections of a hexagonal prism (height 3", IJ" edge), when resting on a face, and an end is 1" from the V.P. 

9. Two prisms (length 3", edge of base 1") are arranged one on the other, so as to form the letter T. Draw the plan and elevation when the front of the letter 
is inclined 25° to the V.P. 

10. An inch square hole, 1" deep, is made in the centre of a square slab (thickness li", diameter 2 J"), and a square prism (length 2i", edge of base 1") is 
placed in the hole. The sides of the hole are parallel to the sides of the slab. Draw plan and elevation when a diagonal of the base of the slab is parallel to 
the V.P. 

11. Show, by its projections a square pyramid (height 3", edge of base 2"), when its base is on the V.P., and an edge of the base is inclined 35°. 

12. Two square prisms (height 2J", diameter 1") are placed one on the other, so that they form the letter T. They tilt over a little, the base of the vertical 
one being inclined 15°. Draw their projections. 

13. Draw the plan and elevation of a hollow cylinder (height 3", diameter 2", diameter of hole 1"). Place it with its axis horizontal. 

14. A cylinder, whose height is 2", and diameter 1", supports a square slab, whose thickness is 1" and diameter 2". Draw their projeotions when the axes 
are both vertical and in the same straight line. 

15. A hollow cylinder (height 3", diameter 2", diameter of hole 1") stands on the H.P., and supports a sphere of 2j" diameter. Draw projeotions. 
Note. — A small circle of the sphere will rest on the top edge of the hole. 

16. A hollow sphere (diameter 3", diameter of hole 2") rests on the ground, and is cut by a horizontal plane 2" high. Draw the true shape of the section. 

17. A vertical hexagonal prism is pierced by a 1 J" cylindrical hole in the direction of its length, and is cut by a plane bisecting it, and making an angle at 45° 
with its axis. Draw the true shape of the section. Height of prism 3J", edge of base 1 J". 

18. Draw plan and elevation of a tetrahedron of 2" edge, when an edge of a face is inclined 45° to the V.P. 

19. A hexagonal prism (height 1", edge of base Ih") supports a tetrahedron, three of whose corners rest on three of the top corners of the prism. Draw plan 
and elevation. 

20. Draw the projections of a tetrahedron of 2" edge, when a face is in contact with the V.P., and the elevation of one of the edges not on the V.P. is 
inclined 45°. 

21. An octagonal slab, whose thickness is 1", supports a square prism (height 2", edge of base IJ"), each bottom corner of the prism resting on a top corner of 
the slab. Draw plan and elevation when a diagonal of the prism is parallel to the V.P. Also show the true shape of a section made by a plane cutting through 
the top left-hand corner of the prism and the bottom right-hand corner of the slab. 

22. Six square prisms (length 3", edge of base 1") are arranged horizontally in such a way that three of them, whose sides are in contact, support two also 



110 INDUSTRIAL DRAWING AND GEOMETRY 

in contact, and the remaining one is placed on top of the two, so that the six together form three equal steps. Draw their plan, and an elevation on a V.P., making 
30° with the long horizontal edges.' 

23. Draw plan and elevation of a tetrahedron of 2" edge, when two of its faces are vertical. 

Note. — If two or more faces or planes meet in an edge or line, they will be vertical if the edge or line he vertical. 

24. Draw the projections of two spheres, diameters 2J" and 1", placing them in contact with one another. Show on the projections where they touch one 
another. 

25. A 2" cylinder rests on the ground with its 2}" axis vertical and If" from the V.P. Show an IJ" sphere touching it and the plane of projection. 

26. A cone, base 2" diameter, axis 3", rests with its base on the H.P., and its axis IJ" from the V.P. An 1|" sphere resting on the groimd touches it. Draw 
their projections. 

27. A cone with its base on the H.P. is out by a vertical plane, the distances of the plane from the axis being J", diameter of base 2^", axis 3". Draw the true 
shape of the section, and give its geometrical name. 

28. A cone, base 2|", axis 3", stands with its axis vertical. Draw the plan and true shape of a section made by a plane bisecting the axis, and inclined 60° to 
it. What is the geometrical name of this section ? 

29. A cone, base 2i", axis 3", is cut by a plane in a direction parallel to its side, and J" from that part. Draw the true shape of the section, and state the 
geometrical name of the curve. 

30. The shape of a slab is a kite- whose diagonals are 3" and If", an angle between two of its sides being 30°, and its thickness |". Draw its projections 
when one of its sides is resting on the H.P., and its plane is 1" from the V.P. 

31. Draw the plan and elevation of a square prism when its faces are inclined 45° to the H.P., and its ends make 45° with the V.P. Edge of base li", 
axis 2i". 

32. Draw the projections of a square pyramid when its axis is horizontal and inclined 40° to the V.P. Edge of base 2", axis 2J". 

33. Draw plan and elevation of a square pyramid when its base its vertical, and one of its faces is parallel to the vertical plane. Edge of base 2", axis 2|". 

34. A face of a tetrahedron of 3" edge is in the V.P., and an edge of that face is vertical. Draw its projections, and show two auxiliary elevations, made on 
planes paraDel to the plan of a face and the plan of an edge. 

35. Draw plan and elevation of a cylinder whose axis is horizontal, and draw a new elevation on a plane whose XY makes 80° with the plan of the axis. 
Diameter of base 2", length 3i". 

36. Draw projections of a cone when its axis is horizontal and inclined 30° to the V.P. Diameter of base 2J", height 4". 

37. A square prism, resting on one of its short edges, has its axis inclined 60°. Draw plan and elevation, and make a new elevation on a plane inclined 30° to 
one of its sides. Edge of base 2", height 3". 

38. A square slab, with a round hole through its centre, is so placed that its square faces are inclined 30° to the V.P., and all its long edges are inclined 45°. 
Draw its projections. Edge of square 3", diameter of hole IJ", thickness 1". 

39. Draw plan and elevation of a square pyramid with one of its faces resting on the H.P. Edge of base 2", height 3". 

40. Draw projections of hexagonal pyramid when one of its slant edges is vertical. Edge of base 1'5", axis 3". 

41. A triangular pyramid rests with its apex on the ground and a face vertical. Draw projections. Base equilateral triangle of 2J" edge, height 4". 

42. Draw the plan and elevation of a square prism, base 2", axis 4", when it is suspended from a point on a long edge 1" from a corner. 

Note. — The line passing through the given point and the centre of gravity of the prism (which in this solid is at the centre of its axis) must be vertical. 

43. Draw the projections of a rectangular block 3" X 2" x 1", when one of its diagonals is vertical. 

' If their long edges be inclined 30° to the V.P., the elevation will be drawn on a V.P. making 30° with those edges. 

= A quadrilateral which has a diagonal as an axis of symmetry has been called by Professor Sylvester a kite. Eefer to Pig. 194. 



CHAPTER XI K 



FURTHER STUDIES IN PROJECTION 



195. First-angle versus Third-angle Projection, or English versics American Practice. — The expedieat we employed in Fig. 290 
ill getting clear ideas about the relation of plan to elevation may be now carried a step further. The first glance at that figure 
might suggest that the horizontal and vertical planes meet or intersect in the XY and go no further; bat in Fig. ;S26 we have 
a pictorial view of a model ^ which clearly shows that the planes cross one another, indeed, we always suppose that they extend 




p 



N- 



REAR 



H 



/ 



b'^ 






^ 



V 



y 

FRONT 



a 



I^JAnqle 



'A 



-<s< 



H 



-^aJ 



^'^."A'/vece 



-< <« 



"77777777^777777/ 



-Y 



--i 



P.I 



Fig. 326.- 



-Piotorial view of planes of 
projection. 



Fig. 328. — Third-angle projec- 
tions of point B. 



Fig. 327. — End view of planes 
of projection. 



Fig. 329.— 
tion 



(t 



First-angle projec- 
of point A. 



indefinitely in each direction. ISTow, Fig. 327 shows an end view of these planes, and we see that their intersection in XY forms four 

angles. That above the H.P. and in front of the V.P. being called the first angle, and thus we have — 

' This model can he readily made by cutting two cards, each through half its length, and halving them together. 

Ill 



■112 



INDUSTRIAL DRAWING AND GEOMETRY 



The first Angle. Above H.P. and in front of V.P, 
The second Angle. Above H.P. and behind V.P. 
The third Angle. Below H.P. and behind V.P. 
The fourth Angle. Below H.P. and in front of V.P. 

The point A (Figs. 32G and 327), we see, is in the iirst angle, and its ordinary plan and elevation is shown in Fig. 329, with its 
plan a (or view from top) below the elevation. And this represents the practice usually followed by English draughtsmen. 

Now, in Figs. 326 and 327, the point B is seen to be in the third angle, and looking in the direction of the arrow M (Fig. 326), 
its elevation will appear at V , below the ground, and therefore below the ground-line or XY in Fig. 328. Looking down on B 
(Fig 326) in the direction of arrow N" for its plan, we see this will appear at 6, behind the vertical plane, and therefore above 
the XY, as in Fig. 328. Thus, when the body is supposed to be placed in the third angle we have the plan above the elevation on 
our paper, and this represents the universal practice of American draughtsmen. And all Americans are extremely sensitive when 
the convenience of this practice is questioned. This being so, it is perfectly certain that they are never likely to fall into line with 
the first-angle projection, which we have seen is generally adopted iu this country. On the other hand, as a large amount of 
American machinery is used in this country, and drawings relating to it, made in third-angle projection, are often being received, 
and, further, as we have many American engineering books used in England in which, of course, things are shown projected in 
accordance with American practice, it is well to become familiar with the system. So, mainly for the above reasons, occasional 
examples in third-angle projectioa are given in the following chapters. Therefore, when you find the plan of a body is shown 
above its elevation, instead of below, as is usual with us, you will know it is American practice, even if it is not marked " Amer. 
Proj." 

196. End Elevations or Side Views. — Look at the three views of a flanged taper pipe shown in Figs. 330 to 332, and those 

shown in Figs. 333 to 335. Which arrangement strikes you as being the 
most convenient ? You will say at once, the former. For, otherwise, ob- 
viously, if the pipe happened to be very long, stretching across the paper, 
the two views, say of the square flange, would be at opposite ends (as in Figs. 
334, 335), a considerable distance apart, and could not be easily compared. 
To avoid this, every end elevation or side view should be shown as the near 
side of the adjacent view from which it is projected,^ as in the upper drawings. 
To get clear ideas about this, ask yourself how you would arrange the 
end views of a long bolt with a square head and a hexagonal nut. You 
would, I think, instinctively place the end elevation of the square head close 
to its elevation, and that of the nut near its elevation. 

Before proceeding to draw projections of some bodies arranged with the 
object of being progressive, you may learn something more about the ex- 
pedients employed in mechanical drawing, by setting out such a simple figure as Fig. 336, which is formed by a combination of lines 

' Unfortunately, there is a want of uniformity of practice in this connection in this country, and the author must plead guilty to having set a bad example in 
some of his early work iu this respect. The reasons for so doing need not be discussed here, but rather should efiorts be made to secure uniformity of practice. 



■°r 


^^1 


i 


X 


Pig. 


330. 


\r 


"f 


[i 


i 



M* 


i n 




j — u 




Pig. 331.- 



-Arrangemeut of end views 
recommended. 



Pig. 832. 




Fig. 333. 



Pig. 33i. — Inconvenient arrangement 
of end views often seen. 



Pig. 335. 



FURTHER STUDIES IN PROJECTION 



113 



and axes, and may be considered an advance on the figures dealt with in Arts. 26 to 28. We will state the case in the form of an 
exercise. Thus — 

197. To draw a Section of a Wrought-iron Beam or Joist. — Fig. 336 is a finished drawing of the section of the beam, drawn 
in a conventional way, and fully dimensioned. After studying the previous exercises, each step the student should take in making 
tliis simple drawing should be obvious ; indeed, all that he should require is a hint or two to enable him to go about it in a 
workmanlike way. Now, this section being symmetrical about a centre line, this line, al (Fig. 337), sliould obviously be drawn 
first, and the rectangular outline of the section, drawn as shown in the figure, forms the first step. It will be noticed tliat the only 
lines in this figure that can be drawn in a finished state right off are AB and C D. The second step is to describe the arcs,^ having 
previously found their centres as indicated at c and d, Fig. 338 (these centres can, with ordinary care and a little practice, be 
found by trial). All that now remains to be done is to carefully join the arcs and complete the outline with lines of uniform thick- 



B 



— radius 
o 




"{ — 


a 

1 

1 








1 


u 


b 



c 



"^ 



c 



+ + 
1 — y 



Y 



JJ 



SECTIONAL 
WEIGHT PER F; 



3 



d 



Pig. 336.— W. I. beam. 
Finished section. 



Fig. 337.— Section of 
beam. First step. 



Fig. 338.— Section of 
beam. Second step. 




Fig. 339. — Section of maximum 
size standard beam. 



ness throughout. The figure may now be cross-hatched or section-lined. The conventional lines in this case (as the material is 
wrought iron)_are alternately thick and thin as shown in Fig. 524; but these beams are now usually made of steel. 

198. British Standard Beam Sections. — The form given to the beam section in Fig. 336 is conventional, it being a convenient 
one for drawing purposes. Formerly there was a great want of uniformity in the rektive thickness of flanges and web, and also 
of the radii of the fillets and edges, to say nothing of the amount of taper given in the flanges ; but in 1904 the Engineering Standards 
Committee published their report on the Properties of British Standard Sections, in which all the sections commonly used by ship 
and bridge builders, etc., are standardized. Fig. 339 gives the standard dimensions for the largest beam section made, which is 
shown here as an example of a standardized section : and it may be used as an instructive drawing exercise. 

' It will be noticed that the ladius of the arcs is three-fourths the thickness of the metal. But it should be explained that these sections are now etandardized 
and the actual radu fixed for all sizes, the flanges being made slightly taper in thickness (as shown in Fig. 339) ; but for some drawing purposes the above 
proportions may be used, and the flanges made of uniform thickness. 

I 



114 



INDUSTRIAL DRAWING AND GEOMETRY 




M I 



- /a 



T 



i 




Fig. 344. — Elevation of stuffing box gland. 



Pigs. 340, 341.— Hook bolt 
(" Amer. Proj."), plan over 
elevation. 



Pig. 342.— 
End elevation. 




-Auxiliary plan to shov^ method of 
drawing. 



looking in the direction of the arrow M. 

200. To draw a Stuflftng Box Gland. 
Scale full size. — Figs. 345 and 344 show, in 
plan and elevation, a gun-metal stuflBng box 
gland (fully dimensioned) for a 2^" piston 
rod or valve spindle/ used to hold the 
packing in the box to keep the joint steam 
tight. 

In commencing a drawing of these 
views, you will first set out the centre lines 
ah and cd, as the object is symmetrical 
about these lines. Now, as matter of prac- 
tice, as has been previously explained, 
whenever one of two views of a body or 
view should be drawn first. So mark out 



Pig. 845.— Plan of stuffing box gland. 





r 












1 
1 




^ 


1 


r^ 


1 
1 
1 
1 


'^ 


# 


1 


^ 


1 






1 

1 









T 

i 



199. — To draw Three Views of a Hook Bolt. — This simple piece, the head part of a hook bolt, has been selected as a suitable 
exercise at this stage in projecting one view from another. In starting such a drawing, it is best, as a general rule, to draw the 

circles first, if there are any. So set out 
the end elevation (Fig. 342). You will now 
experience no difficulty in doing this. Next 
draw the elevation (Fig. 341), being careful 
to make the arc of 1" radius flow into the 
straight lines, and complete the exercise by 
drawing the plan (Fig. 340), which in this 
case is shown above the elevation (" Amer. 
Proj."), but if you prefer it draw it below. 
You will notice that the end view (Fig. 
342) is got by looking at the bolt in the 
direction of the arrow N, and the plan by 





+ 


(^ 




5 1 




/'. 




^1* 
















1 

1 




^ 






Plan 





< 


- ^4 


c 


> 




1 
1 


1 1 
1 1 


1 1 


1 
1 


1 


P 


i# 




— L 


1 

-HOO 














* 3^ 








part of a body is circular in form, that 

centre lines for the holes A and B (as in Fig. 343), and describe the four circles in 

plan to the dimensions shown, giving the lines their finished thickness. Then, with 

1" radius, arcs may be drawn about the centres of the stud holes A and B with a light ^^^- 346.— Section on line ah. 

line, also arc DJ, of 2-^^!' radius, about centre K; then tangents such as CD can be drawn, and the plan completed (as in Fig. 344) 

by going over the arcs EF and GH, etc., making them uniform in thickness, with the other lines. The elevation presents no 

difficulty, and should be easily drawn now. 

' For particulars relatiug to stuffing boxes, etc., see Arts, from 63, Author's " Machine Design, Construction, and Drawing for Beginners." 



FURTHER STUDIES IN PROJECTION 



115 



"777777^, 



P'^ 



At this -stage, a good exercise on the above would be to draw a section of the gland made hy a plane cutting it in halves through 
the line ah (Fig. 345). Obviously, its outline would be similar to the election. Fig. .344, and Fig. 346 shows the finished section. 
Such a gland would be made of brass, and the conventional section-lining (as in Fig. 524) for this metal has been used. 

201. To draw the Plan and Elevation of a Steel Crank, also a Section on the line AB (Figs. 347 to 349). — Have a good 
look at the projections of the crank, and you will agree that the elevation (which, as there are circles on it, you will draw first) must 
be drawn with great care, if the tangential arcs forming the outline are to 
meet properly. If you are in any doubt as to how the centres of the two 
arcs of 500 mm. radius are to be found, turn back to Art. 75, and Fig. 101 
will help you. For, obviously, the centres of the arcs forming the sides, the 
radius of which is 500 mm., will be (500 - 135) mm. from C, and (500 - 75) 
mm. from C3. The outline being drawn, the remaining part of the elevation 
should present no difficulty, and the plan is readily projected from the 
elevation. 

Coining to the section : if the part ADB be cut away by the section 
plane, then, looking in the direction of the arrow M, we get the view shown 
in Fig. 349. Section-lining the parts that would be cut through by the plane, 
with dotted lines, as in Fig. 524, to represent the material, steel. 

202. Use of Pictorial Sketches and Drawings. — Look at Fig. 292, which 
is a pictorial drawing or sketch. You see at a glance the shape of the block 
of wood represented, also its position in relation to the planes of projection, 
as the drawing has the advantage of conveying in one view ideas of the 
three dimensions. The two views of the same block shown in Fig. 293 do 
not so readily convey to your mind the same information. Also have a look 
at Figs. 299 to 304, the pictorial sketches shown are dimensioned. If you 
were sent somewhere to take particulars of one of these pieces, it would 
be convenient to make such a freehand sketch, and after carefully measuring 
the piece with your rule, write on it the dimensions ; you could then, on your return, make a proper drawing in plan and elevation 
of the piece. You will also notice that the front face of each of the pieces shown in the figures is seen in true shape. But if you 
scrutinize Fig. 296 you will notice that no face is shown in true shape ; as a matter of fact, it is an isometric ^ drawing, a view 
of the block tipped up into such a position that all its edges are equally inclined to the ground, and therefore equally foreshortened in 
plan. To enable you to make such drawings and sketches with facility, we will devote the next two chapters to their consideration. 

' Gr. from isos, equal, and metrrni, a measure. This system, of drawing was introduced by Professor Sir George Stokes of Cambridge. 



'7777777777} 




135 — >!«- 



300" 



H<- 7.9- 



PiG. 349.— Section 
on line AB. 



Figs. 347, 848.— Elevation and plan of steel 
crank (" Amer.Proj."), plan above elevation. 



CHAPTER XX 

ISOMETRICAL PROJECTIONS AND DRAWINGS 



203. Introduction. — You have read the previous article, and will understand that if the cube shown in Fig. 350 be cut by 

a plane passing through the corners h, f, d, the three face diagonals hf, fd, and db, being the same length, form an equilateral 

triangle, to the plane of which the three equal edges cb, cd, cf are equally inclined. 

In Fig. 351 the orthographic views P and N", the plan and elevation (third-angle projections, Art. 195) of the cube are shown, 

and you will notice that the solid is so placed that the face diagonal a'c' is paralled to the V.P. Therefore h'c' (view ¥) is a 

diagonal of the cube seen in true length. Further, the cutting plane we have 
referred to in connection with Fig. 350 is shown here by the line i'f, passing 
through the three corners h', d', /', and perpendicular to the diagonal c'h'. The 
new plan E, due to looking at the cube in the direction of this diagonal (or 
due to the diagonal being vertical) is shown, and you will experience no 
trouble in drawing it, after having studied Fig. 316, Art. 192. You won't 
fail to notice that the boundary line is a regular hexagon, that Ci is the centre 
of the triangle lidifi, which is equilateral, therefore the three angles at ci 
are each equal to 120° ; and that every other edge of the cube is equal and 
parallel to one of those meeting in the centre ci, and the plan of each face a 
rhombus. The plan E is called an Isometric Projection because all the edges 
of the cube are foreshortened the same amount, and therefore all lines parallel 
to them can be measured with the same scale. 

The three edges ci&i, Cidi, ci/i, are called the Isometric Axes, and the 
planes which they determine (that is, there are three such planes, each con- 
taining two of the axes), and all planes parallel to them, are called Isometric 

Planes, whilst all lines parallel to the axes are called Isometric Lines. 

204. The Difference between Isometric Projection and Isometric Drawing. — Look at view N, Fig. 351. You will see that c'f 

is the true length of the cube's edge, whilst in the new plan E the line Ci/i is its apparent length.^ Now, obviously, if c'f is 

made 1 inch in length, Cj/i will be what we may call an isometric inch. And if these lines are produced, inches can be laid off 

' You ought to be able to satisfy yourself that the ratio of the apparent length to the true length is fJ9, ; v'3. 

116 



m 




EiG. 350.— Pictorial 
drawing of cube. 



Pig. 351. — Isometric 
projection of cube. 



E F 

Pig. 352. — Isometric 
drawing of cube. 



ISOMETRICAL PROJECTIONS AND DRAWINGS 



117 



from fc' produced, and their projections on /iCj produced will give an isometric scale, which can be used in constructing any 
isometric projection. Now, this question of the scale is purely an academical one, as it has no practical value, for since the 
isometric lines are all equally foreshortened, there is no reason why they should represented as foreshortened at all. Therefore 
in practice an isometric drawing (Fig. 352) of our cube would ordinarily be made by drawing each edge its true length, 
dimensions being set off from an ordinary rule or scale. Obviously, drawings made in this way have the great advantage 
that the ordinary rule or scales can be applied to them by a workman to measure any part. 

Of course an isometric drawing of a body will be larger than its isometric projection in the proportion of Fig. 352 to the 
projection E (plan) in Fig. 351. And it can be shown that the corresponding lines are in the ratio of \/3 to \/2. 

c 




a 

Fig. 353. — Isometric axes, ah and ad. 





Pig. 354. — Isometric drawing of a cube. 



Pig. 355. — Isometric drawing of a brick. 



205. Some Typical Isometric Drawings. — Of course, to make an isometric drawing of a rectangular body, such as the cube, 
direct, all we have to do is to draw lines ah and ad, from any point «, Fig. 353, in a line XY, inclined 30° in opposite directions' 
and measure the length and breadth of the body along them, and the thick- 
ness along a line ac perpendicular to XY ; the view of the solid is then easily 
completed by drawing par<^Dels to these lines, as shown in Fig. 354. The 
block or brick shown in Fig. 290 is drawn in this way in Fig. 355, and such 
drawings should now speak for themselves. 

In Figs. 356, 357 you have the orthographic projections (" Amer. Proj.") of 
an iron shed sho^va, and in Fig. 358 an isometric drawing of it. The only 
point about the latter that will trouble you when you attempt to make the 
drawing is the positions of the points A and D, where the slant hip-rafters 
meet. But you will, no doubt, see that the offsets BC and CA, in Fig. 358, 
must be made equal to Vc and da in Fig. 357 respectively. 

206. Isometric Drawing of a Circle. — If a circle be circumscribed by a 
square, as in Fig. 359, and the diagonals drawn, these cut the circle in a, c, e,g, 
through which lines parallel to the sides enable you, in making the isometric 
drawing (Fig. 360), to find similar points in the ellipse, whose axes coincide with the diagonals, as shown in the figure. The curve 
can be carefully drawn through the eight points so determined ; or additional points can be used (and transferred) as shown at the 




■■'///MM///)////////////////////, 
Pigs. 356, 357.— Elevation. 



Fig. 358. — Isometric drawing. 



118 



INDUSTRIAL DRAWING AND GEOMETRY 



intersection of the dotted lines (Fig 359). Another method of finding points in the ellipse is also shown in Fig. 360. On the 
diameter ac of the circle describe the semicircle aWDe, and by setting off angles at 45° and 30° from the centre, the former gives 






Fig. 359. — Circle ciroumsoribed by square. 



Fig. 360. — Isometric drawing of circle 
with the circumscribing isometric squares. 



Fig. 361. — Isometric drawing of a circle. Another method. 



B and D, from which perpendiculars to ae cut the latter in h and d ; whilst the angles of 30° intersect in c ; and the remaining 
points (7, h,fin the ellipse may be found by symmetry. A variation of this method, shown in Fig. 361, should almost speak for 
itself. A semicircle ced is described on one of the sides of the rhombus, and divided into four equal parts by /, e, and g. The 
perpendiculars fi, ej, and gh to the side cd give the points i, j, and h, through which the parallels are drawn, the cross ones being 
found in the same way or by symmetry ; the intersections giving four points in the ellipse, and the centres of the sides of the 
rhombus the other four. 

207. Setting off Angles to the Sides of the Isometric Cube. — Let Fig. 363 be the isometric cube, then draw a square, whose side 
ah (Fig. 362) is equal to the edge of the cube. With one of its corners, say a, as centre, describe the quadrant bd, divide it into 

the required angles, and produce the radial lines through the points of division to cut the 
sides of the square, as in e, f, c, g, forming a scale of tangents. As an illustration of the use 
of this scale : with the compasses prick off h'e, Fig. 363, making it equal to he (Fig. 362) ; 
join a! to «', and this is the isometric projection of a line which makes an angle with the edge 
ah of the cube (Fig. 362). You will see that two other angles have also been set off on Fig. 
363 by making a'f" and d"'g"' respectively equal to hf and dg on Fig. 362. 

Other expedients could be shown you, some of which you may discover for yourself, if 

you take any real interest in this system of projection ; but, as we have seen, the oblique or 

pictorial system of projection, which enables us to show one face of a body in true shape, is 

so much more used for practical purposes that your time will be better spent in becoming 

angles by" using^ the ™ore familiar with it, as you will be if you carefully follow the next chapter. However, 

scale of tangents. before doing so, you might with advantage work the following exercises. 




Fig. 362.— Scale 
of tangents. 



ISOMETRICAL PROJECTIONS AND DRAWINGS 119 

EXERCISES. 

TspicAL Oeal Exbhcisbs. 

1. What is the meaning of the term " isometric " when applied to a projection ? 

2. What is the diiierenoe between isometric projecticms and isometric drawings ? 

3. How many isometric axes has the isometric projection of a cube ? 

Dbawing Exeecisbs. 

4. Make isometric drawings of the solids shown in Pigs. 300, 801, and 302. 

5. Make an isometric drawing of a square pyramid, edge of base If", axis 2J". 

6. Make an isometric drawing of a 3" circle. 

7. A cylinder is 2" in diameter and 3" in length. Make an isometric drawing of it. 

8. Make an isometric drawing of your instrument box, with the lid upright. 

9. A square slab 2i" across and 1" thick has a 2" hole bored through its centre. Make an isometric drawing of it. 

10. Ask your teacher to lend you the chalk-box. Carefully measure it, and make an isometric drawing of it when the lid is in some inclined positicn, say, 
making an angle of 45° with the top of the box. 

11. A 2" cube is cut by a plane passing through one of its edges and making an angle of 30° with a face. Make isometric drawings of the two parts when 
separated. 

12. Make an isometric drawing of the shed shown in Fig. 802. Scale \" to the foot. 

13. Make isometric drawings of the blocks shown in Pigs. 303 and 304. 



CHAPTER XXI 



OBLIQUE OR PICTORIAL DRAWING 

208. Introduction. — We have already made use of a system of drawing that lends itself to the rapid representation of a body 
showing its three dimensions on one view, and have referred to it in Art. 202, which yon might again glance through. By this 

system, known as oblique projection, all lines lying in planes parallel to 
the paper are shown in their true forms, lengths, and relations. Further, 
all lines perpendicular to the paper are shown in true length or in some 
suitable fraction of that length. It is a system well adapted to some of 
the requirements of many trades, particularly those of the carpenter and 
joiner, where difficult joints and fitted pieces have to be shown. 

The four different drawings of the same perforated prism, in Figs. 
364 to 367, are arranged to show that the edges, which are perpendicular 
Pigs. 364 to 367.-Pour pictorial views of a perforated prism. ^^ ^-j^^ p^pgr, may be represented by paraUel lines having any suitable 

direction ; this enables you to show, not only the front face, but either top or bottom, or the right or left side of a body, and either 
of these faces can be made more noticeable than the other by care in the arrangement of the angle. Of course, it is a convenience 



- 


^ 


o 

k 




-^ 


1 


\ 




{^, 





Pig. 368.— Block of steps. 



Pigs. 369, 370.— Notched joint. 
120 




Figs. 371, 372.— Co, 



OBLIQUE OR PICTORIAL DRAWING 



121 



to select one of the set-sqnare angles. Thus it will be seen that 45° has been used in Fig. 364, 30° in Fig. 365, and 60° in Fig. 
366, whilst in Fig. 367 it is IS''. Circles on planes perpendicular to the paper project as ellipses, and the rules apply as 
explained in Art. 206. 




Figs. 373, 374. — Grooved and tongued joint. 





Common dovetail joint. 



Pigs. 375, 376.— Showing the two 
parts separated. 



Pig. 377.— Showing the complete 
joint. 






.£-—- 







Pigs. 378, 379. — Mortise and tenon joint. Fig. 380.— Mortise and tenon joint put together. 



Pig. 381. — Housemaid's blacklead bos. 



209. Some Simple Bodies drawn in Oblique Projection. — Most of the examples shown in Figs. 368 to 387 will practically speak 
for tliemselves. Some of the joints most commonly used in woodwork are shown. The picture-frame mitred joint (Figs. 382, 383) 



V, 



INDUSTRIAL DRAWING AND GEOMETRY 



is an example which shows that there are certain limitations both in this and in isometric projections beyond which the systems 
should not be pushed.^ It will be noticed that the apparent breadth x of the side of the frame comes out in the drawing a good deal 
wider than the real breadth ; indeed, this is an example which would have come out better in an isometric drawing. Of course, the 
method of using offsets employed in connection with Fig. 358 is applicable here for the mitre c'd' in Fig. 382, being equal to cd in 
Fig. 383, also a'b' is made equal to ab. Fig. 384 is an oblique projection of the cast-iron bearing block shown in Figs. 471, 472. 



DRAWING EXERCISES. 



1. Draw a pictorial view of tlie stationery case (Fig. 385). Scale h size. 

2. Make drawing in oblique projection of the sign-post in Figs. 386, 387. Scale J size. 

3. Make a drawing in oblique projection of the hook-bolt in Figs. 840 to 342. 
Draw in oblique or pictorial projection the following: — 

4. The notched joint (Figs. 369, 370). First draw the parts separated, as shown, then draw the complete joint. 



Scale J full size. 




&' 


■^ ' 


X 




—rr^ 


\a 




Figs. 382, 383. — Picture-frame mitred joint. 



Fig. 384. — Cast-iron bearing block. 
Showing parts in section. 



5. The cogged joint (Figs. 371, 372). First draw the parts separated, as shown, then draw the complete joint. Scale J full size. 

6. The grooved and tongued joint (Figs. 373, 374). First draw the parts separated, as shown, then draw the complete joint. Full size. 

7. The common dovetail joint (Figs. 375 to 377). First draw the parts separated, as shown, then draw the complete joint. Full size. 

8. The mortise and tenon joint (Pigs. 378 to 380). First draw the parts separated, as shown, then draw the complete joint. Scale | full size. 



' Many attempts have been made to draw a general plan of a complicated machine in isometric or oblique projection, but the contortions, which are hardly 
noticed in the case of small details, become so pronounced as to make the system quite useless for such purposes. 



OBLIQUE OR PICTORIAL DRAWING 



123 



9. The housemaid's blacklead bos (Pig. 381). Scale i full size. 

10. The picture-frame mitred joint (Figs. 382, 388). Pull size. 

11. The bearing block, sectional view, as in Pig. 881. Scale double the size shown. 

12. A 8" square slab, J" thick, supporting at its centre an li" sphere. 
18. A cone, base 2", axis 8", resting on a 3" square slab, |" thick. 



« S" 



«)l* 
■? 




'yy/y/y/>y^yy/y^/'/'/yyy////i'////'A 



"hickness all over | " 

Pig. 385. — Stationery case. 



^^^ 



Pigs. 386, 387.— Sign-pds^ 



CHAPTER XXII 

SIMPLE FASTENINGS USED IN METAL WORK, AND HOW TO DRAW THEM 



Riveted Joints 

210. One of the most simple and efBcient fastenings, which has been extensively used for a great variety of purposes from very 
ancient times, is the rivet. As a fastening, it somewhat resembles a bolt, but differs from it in two important respects ; for a bolt 
can be used as a temporary fastening, and can be withdrawn by unscrewing the nut ; but a rivet is a permanent fastening, and the 
parts held together by it can only be separated by chipping off a head. Further, a bolt is used satisfactorily when the straining 
force acts in the direction of its axis, giving it a tensional load, but it is not considered advisable to load a rivet in this way, its 
proper function being to resist shearing in a direction normal to its axis. 

Eivets are made in special machines (from special round iron or steel bar), with heads either cup-shaped, as in Fig. 388, or 
pan-shapsd, as in Fig. 389 ; the heads are formed while red hot by dies of these shapes, and their finished forms before and after 
use are shown in the figures ; the dotted lines showing the length of the stub end required to form the second head. 

Unit D-Diam.of Rivet Hole 

' , «-' 





Pig. 388. — Head and second 
head cup-shaped. 



-I-4-6- 



Fig. 389. — Pan-shaped head, second 
head fully countersunk. 





Pig. 390.— Conoidal second 
head, hammer-finished. 



Pig. 391. — Proportions for 
drawing purposes. 



In riveting plates, whenever practicable riveting machines are used, the rivet is made red hot, passed through the plates and pressed between 
two dies, by hydraulic or steam pressure. The heads are then usually made cup or spherical shaped, as in Fig. 388, and are said to be machine 

124 



SIMPLE FASTENINGS USED IN METAL WORK, AND HOW TO DRAW THEM 



125 



riveted. Wlien machines are not available the rivets are hand riveted. For this job a full gang consists of three men and a boy, the latter to heat 
the rivet and bring it from the fui-naee to the holder up, who inserts it into the rivet hole and presses against the rivet with a tool caUed a dolly, cupped 
to receive the head of the rivet, while the other two men on the opposite side hammer the stub end down with riveting hammers and finish it off by a 
blow or two from a sledge hammer, a snap-headed tool being interposed to give the head the cup shape in Fig. 388. In confined positions where it is 
not possible to snap the heads, they are finished by hammering to the conical or conoidal form shovm in Fig. 390, which has usually not quite the 
sti-ength of the cup head. In many classes of work, such as the plating of ships, the seatiugs of girders, etc., the heads must not project ; the plates 
are then countersunk, as shown in Fig. 389 (which shows a fidl counter-simk head), and the heads finished off flush with the plate, or with a slight 
fulness or projection, as shown dotted. 

For drawing purposes an approximation to the ordinary cup head is easily made by using a radius of | the diameter, as shewn 

in Fig. 391, and striking the head from a point on the centre line \T) from the shoulder. 

211. Proportions of Rivet Heads, etc. — These proportions vary somewhat in practice, as 
they have not yet been standardized ; but those shown in Figs. 388 to 390 may be taken to be 
average ones, they are in terms of D, the diameter of the hole. The dotted lengths for forming 
the heads should be taken to be approximate. They vary from 1'25 to 1'7 times the diameter, 




SECTION ON RIVET LINE. 



.^±x 



- t^^M^^r j$^ ?2m 








■^''t^'^l 





Figs. 392, 393.— Single-riveted lap joint. Figs. 394, 395.— Double-riveted 

lap joiut (chain). 



Figs. 396, 397.— Double- 
riveted lap joint (zigzag). 



Figs. 398, 399.— Butt joint with double 
butt straps or cover plates (zigzag). 



the actual length required depending upon the completeness with which the rivet fills the hole, and upon whether the head is 



126 INDUSTRIAL DRAWING AND GEOMETRY 

formed by hand or by machine, the former requires about |D less length than the latter, as the machine compresses and swells the 
rivet till it completely fills the hole, thus making a very perfect job. 

212. Forms of Joints. — The simplest form of riveted joint is the lap joint, with a single row of rivets, shown in Figs. 392, 393; 
this joint, although largely used for many purposes, has, when subjected to great straining actions (as it is in boiler work), an 
obvious fault, for a couple acts about the rivets, tending to bend the joint (as shown in Fig. 392), owing to the plates A and B not 
being in the same plane. The butt joints, to be directly described, make a much more satisfactory but more expensive job. 

213. Double-riveted Lap Joints are shown in Figs. 394 to 397. In Figs. 396, 397 we have rivets arranged zigzag, and in Figs. 
394, 395 they are opposite to one another, or the joint is said to be chain riveted. 

214. Proportions of Joints. — The usual practice is to make the distance x (Fig. 392) between the side of a rivet and edge of 
the plate (called the margin) at least equal to the rivet diameter, thus making the minimum lap equal to od, as shown, but in cases 
where the edges of the plates are more or less rough, a j" is added to this. 

215. Diameters of Eivets. — These are shown on the drawings, but the diameters of the rivets for other thicknesses of plate may 
be found by using the empirical formula dia. d = l-2\/t, where t is the thickness of the plates. 

216. Pitch of the Rivets.— The minimum pitch in any given case may be determined by the rough formula, pitch = d + 1|". 
In the example. Figs. 392, 393, the pitch, it will be noticed, is 3^. The smaller pitches are used when the joint is to be kept 
steam-tight. When the strength of the joint is to be the greatest possible, the pitch is determined as explained in the author's 
" Machine Design." 

217. Butt Joints with Double Straps. — Figs. 398, 399 show a double-riveted butt joint with douUe butt straps. In this joint, 
the plates to be joined are in the same plane, and they butt one on the other, top and bottom cover plates or straps being placed 
over and under as shown. It has been found by experiments that when the straps are made half the thickness of the plates (as it 
would appear they should be) the straps are then the weakest part. This has led to the practice of making their thickness from g< to t 
(thickness of plate). The dimensions on the figures are suitable for |" plates ; other proportions are shown on the figure. 

The distance between rivet lines for chain riveting is given on Fig. 395. The distance y (Fig. 397) for zigzag riveting may be 
found by the rough rule y = l'7d. 

For further information about riveted joints, refer to the author's " Machine Drawing and Design for Beginners." 

218. Hints on Making the Drawings. — The sections in all the joints shown should be drawn first, commencing with the centre 
lines through the rivets, the thickness of the plates can then be marked off on this line, and the plate lines drawn. The rivet 
should be next drawn. Referring to Fig. 391 for its details, you will see that the radius of the head, for ordinary drawing purposes, 
may be | of D ; then, as D = f ", the radius is f x f " = f'g", and the centre is J X |" = ^\" below the shoulder, so, with radius 
off", and this position of the centre, form the heads, and complete by section- lining, as shown, the upper view; the plan can then 
be readily projected from the section. Scale for all the views, full size. 

219. Screws, Bolts, etc. — It will now be convenient to give some attention to the pair of elements forming the fastening, which 
in the science of kinematics ^ is called a screw-pair, the simplest form of which is the common bolt and nut shown in Fig. 407. A 
fundamental feature of bolts and screws is that parts connected by them can be easily disconnected when required, and, when it is 
realized what a great variety of work these interesting fastenings are used for, some idea can be formed of the multiplicity of forms 

' Science of pure motion. 



SIMPLE FASTENINGS USED IN METAL WORK, AND HOW TO DRAW THEM 



127 



and kinds that are in actual use ; but for our purpose we shall only give attention to two or three of the most important ones. 
JSTow, to completely specify some special form of bolt or screw it may be necessary to mention eight features, namely, (a) shape or 
form of the thread, (b) pitch or number of threads to the inch, (c) shape of head, (d) outline of body, barrel or stem, (e) size or 
diameter, (/) direction of threads (as right-hand or left-hand), (g) length, (A) material, as iron, brass, etc. 

220. Forms of Screw Threads. — Figs. 400 to 403 show tlie threads most commonly used by the enn-ineer. Fi"s. 400, 401, a vee 



'-p-i 







Fig. 400.— Whitworth's. 



Fig. 401. — Whitworth screw. 



Fig. 402. — Seller's screw. 



Fig. 403. — Square thread. 



thread, slightly rounded at the top and bottom, is Whitworth's, the standard British thread. Fig. 402 is also a vee, with the top and 
bottom slightly flat, it is Seller's and the standard thread of America ^ Fig. 403 is the square thread, which you have no doubt 
seen on the letter-copying press ; unlike vee threads, this screw 



WHITWORTH'S THREAD 



PITCH— Ji 

Fig. 404.— Detail of the British 
Standard Whitworth thread. 




SELLER'S THREAD 
}•- PITCH — »{ 



does not subject the nut to a bursting strain. 

221. Proportions of Screw Threads. — Fig. 404 shows the shape 
of our Whitworth vee thread. The angle between the threads 
being 55°, and -J-th of the full depth of the triangle abc being 
rounded off at the top and bottom (to a radius of 0137329^), as 
shown. But the full depth of the triangle is 0'96 the pitch, so 
that the actual depth of the thread is | X 0'96j3 = O'Qip, or, to be 
exact, 0-640327^. And if d = diameter of the screw at top of the 
threads, Fig. 401, and di = diameter at bottom of the threads 
(the net or core diameter). 

Then the core diameter di = O'M - 005, nearly (1) 

And if w = number of threads per inch, and p = the pitch of the threads. 




Fig. 405. — Detail of the American 
Standard Seller's thread. 



Then the pitch 



p = - = O-OSd 4- 0-04, nearly 



n 



(2) 



In Fig. 405 is shown Seller's thread, which we have explained is the standard thread adopted by America. The triangle in 
this case is equilateral, the angle therefore being 60°, I the full depth of the triangle being cut off top and bottom, as shown, to 

' It is claimed that the dies and taps used to produce these threads can be used longer before becoming blunt than ours ; but, strangely enough, our 
Whitworth threads are used in the American Navy work. 



128 



INDUSTRIAL DRAWING AND GEOMETRY 



form flats parallel with axis. So that the actual depth of the thread d! = |-rf, or d' = '^ X 0'866^ = 0'65^. The proportions of the 
s(iuare thread (Fig. 403) are shown in Fig. 406 ; the pitch for standard screws being twice that for vee threads the same diameter, 




EQUILATERAL 
TRIANOLCtoOBTAI 
APPROXIMATE DEPTH 
OF THREAD. 



- A- ■ 

Pigs. 407, 408. — Proportions of hexa- 
gonal bolts for drawing purposes. 



or, the pitch for square threads 



P = 



n 



0-Wd + 0-08, nearly 



(3) 



And, if di — diameter at bottom of threads (the core diameter), as in other cases, 

Then the core diameter for square threads di = 85d — 0-015 . , (4) 

With this thread the thrust is very nearly parallel to the 
axis of the screw, and therefore there is uo bursting strain 
on the nut, as we have seen, which is an important advan- 
tage. But the thread is more costly to produce than the 
vee thread, more particularly as it cannot be satisfactorily 
cut with dies. The figure (406) shows the usual propor- 
tions of the thickness and depth of the threads. 

222. Drawing Exercise. To draw an 1" Whitworth 
Bolt and Nut. — From a drawing point of view by far the 
most important detail you will have to deal with is the 
bolt and nut, as any want of accuracy in presenting it 
mars the appearance of what otherwise might be a very 
good drawing, and offends the trained eye. Further, as the 
detail so often occurs on drawings, a real effort should be 
made to set it out in the usual conventional way shown in 
Figs. 407, 408, and 409. 

Commence witli the Plan^ Fig. 408, by drawing the 
circumscribing circle (with a radius ^ equal to d, the dia- 
PiG. 409.— Side eie- Fig. 410.— Bolt with meter of the bolt = 1"), and the bolt circle (radius i"), and 
vation of Fig. 407. square head and nut. ^^^^ ^jjg jg^^^gj. ^^^^, projectors. Cutting the former in a and 

h, join ah, and describe the chamfer circle, touching ah in c. The hexagon is then completed with 
the 60° set-square, making each of the other sides just touch the chamfer circle. Projectors from 
the corners e,f can now be drawn, and these, with projectors from a and &,"give the indefinite 
elevation of the bolt body, and edges of nut and head. The thickness of the nut {= d) can now be 
set off, and with radius l'2c?, and centre on centre line, the arc /K can be drawn, and a line 
through these points gives M and JST, which are used, as shown, to draw the arcs on the side 

' As you have seen, if you were drawing this in accordance with Ameiican practice, you would place the plan above 
the elevation. 

- As we have- explained, for drawing purposes (for 1" holts and under) it is convenient to make the diameter across 
the angles = 2d. 




SIMPLE FASTENINGS USED IN METAL WORK, AND HOW TO DRAW THEM 



129 



faces/ the elevation of the nut is then completed (if you wish to make a very exact drawing) by drawing the chamfers at 30°, to 
just touch the arcs ; but you need not trouble about this chamfer for your purpose. The head is drawn in the same way, making its 
thickness equal to O'dd, whilst the point or end of the bolt is usually rounded with a radius = d. The screw threads are easily drawn 
in the conventional way shown, the slope being fixed by marking up i the pitch ; the thick lines, of course, represent the bottom of 
the threads, and their length may be found by making the small equilateral triangle shown, of side equal to the pitch, which gives 
the approximate depth of the thread. The use of the dotted lines on the nut will be apparent. 

Pig. 410 shows a bolt with square head and nut, and square neck to prevent the bolt rotating whilst screwing up, the bolt hole 
being sqiiare ; the proportions given in the table below, with the exception of the diameter across the angles, also apply to these 
bolts. 

223. Standard Bolts and Screws. — We may now give some attention to the proportions of bolts and bolt heads in general use. 
Figs. 407, 408, and 409 show, as we have seen, the form of the common hexagonal bolt and nut ; their proportions are now 
standardized,^ they are practically those given in the table below, which are in common use. The practice of some manufacturers 
in the past has been to make bright nuts and heads somewhat smaller in diameter than black ones, but this is very inconvenient, as, 
if for no other reason, it necessitates the use of two spanners for the same size bolt. However, as now standardized, both the bright 
and black have the same maximum dimensions, the minimum dimensions fixed for the latter giving a larger allowance, as it is 
called. 

Pkopobtions of Standakd Whitwobth Bolts and Screws . 



Diam. of Bolt 

or Screw. 

Inches. 


No. of Threads 

per inch 

= n. 


Diam. across 
Flats = D. 


niam. across 
angles — 1-155 D. 


Diam. of Bolt 

or Screw. 

Inches. 


No. of Threads 

per inch 

= n. 


Diam. across 
Flats = D. 


Diam. across 
angles = 1-165 D. 


• i 


20 


0-525 


0-6062 


1 


8 


1-6701 


1-9284 


i 


16 


0-7014 


0-8191 


n 


7 


2-0483 


2-3651 


J 


12 


0-9200 


1-0612 


n 


6 


2-4134 


2-7867 




11 


1-101 


1-2713 


li 


5 


2-7578 


3-1844 


f 


10 


1-3012 


1-5024 


2 


4-5 


3-1491 


3-6862 


I 


9 


1-4788 


1-7075 











224. Locking Nuts and Arrangements. — ISTo matter how perfect the fit of a nut on its bolt may be, when it is subjected to 
vibration, or to the jarring tremulous motion of machinery, the nut gradually works loose or tends to do so, and may, if there 
is nothing to stop it, work off the bolt. Now, one of the best known expedients to prevent this, and the one usually employed when' 
pieces subject to rapid motion nre connected by bolts, is the Lock Nut, which is an extra nut screwed tightly down on to 
the ordinary one, as in Fig. 411, to jamb or lock it on the bolt in such a way that it will not work loose. This lock nut is sometimes 
made half the ordinary thickness of a nut, on the assumption that it is only to jamb the other nut and take only a small part (if any) 

' A little practice will enable you to draw these with considerable accuracy and facility by feeling for the centre and radius, assuming tentative radii and 
positions of the centre tiU the true centre is found. 

- Refer to Reports on British Standard Screw Threads, published by Crosby Lookwood & Son, for further information, if required. 

K 



130 



INDUSTRIAL DRAWING AND GEOMETRY 



of the load, but a little consideration will satisfy you that it is the top nut which practically takes the whole load, whatever its 
thickness may be, and, therefore, of course the thick nut should be there (as in Fig. 411), as the true lock nut,i but spanners (or 
wrenches) are rarely thin enough to take the lock nut when it is thin and is placed at the bottom, and this has led to the growth 

of the faulty practice shown in Fig. 412. An obvious way 
out of the diificulty would be to make both nuts the full 
^^ thickness, but there is not always room for this, and when 
there is it offends the eye, so the compromise of keeping the 
total thickness the same and making them both the same 



thickness, namely f to jd, as in Fig. 413, is one that is often 
met with. However, the standard arrangement, shown in 
Fig. 411, should always be used when convenient. Fig. 413 
also shows how the end of the bolt is sometimes turned down 
to allow the nut to be easily screwed on, and to more con- 
veniently allow of a split pin to be used, where the bolt is 
subject to much vibration, to prevent the nuts working off, 
should they get loose. 

225. The Capstan nut or Castle nut (Figs. 414, 415) is 
largely used for locking purposes in motor-car work, and on 
jobs generally that are subjected to sudden shocks and much 
vibration. It consists of an hexagonal nut, with a portion 




Fig. 411. — Standard 
practice. 



Fig. 412.— Practi- 
cally convenient, 
theoretically 
faulty. 



Pig. 413. — Compro- 
mise, sometimes 
convenient. 



Figs. 414, 415.- 
Capstan nut. 



turned off making a circular collar, through which rectangular slots are made, and into which, after the nut has been adjusted, a 
round or rectangular cotter with split ends is fitted through both nut and bolt. The standard proportions are, D = width across flats 

T = l-25d, H = 0-75d, t = 0-4375c?, W = Q-2M, and the radius E may = | 



1 " 

16 I 



On the other hand, a bolt is 



EXERCISES. 

Typical Obai, Exebcisbs. 

1. What advantage has a bolt over a rivet ? 

2. In structures such as girders and boilers the parts are riveted together, as the handle of a frying-pan is riveted to the pan. 
used to attach the bell to your bicycle handle. "Why should not bolts be used for the former cases, and rivets for the latter ? 

3. Why does a butt joint in riveted plates subjected to great tension, make a better mechanical job than a lap joint ? 

4. What advantage has a square-threaded screw over a vee-threaded one ? 

5. How are nuts prevented from working loose or coming ofi when bolts are subjected to considerable vibration ? 

6. What is the difference between the screw threads used in this country and those used in America ? 

' It is the practice of some engineers to arrange the nuts in this way, and to make the thickness of the bottom one equal to d and^the top one equal to Id. 



SIMPLE FASTENINGS USED IN METAL WORK, AND HOW TO DRAW THEM 131 

Sketching Exebcises. 

7. Slake good bold freehand sketches of the following : — Rivet heads, (o) A cup or snap head, (b) A countersunk head, (c) A hammer-finished head. 

8. Make a freehand sketch in good proportion of (a) a single-riveted lap joint ; (b) a double-riveted zigzag butt joint. 

9. Make eiiective freehand sketches of (a) a Whitworth screw thread ; (6) a Seller's screw thread ; (c) a square screw thread. 

10. Show by a freehand sketch in good proportion a standard lock nut arrangement. 

11. Make a careful freehand sketch of a castle lock nut. You had better use an actual nut to work from if you can borrow one from your teacher. 

Db AWING Exebcises. 

12. Make a neat drawing of the lap joint shown in Figs. 392, 393. Pull size. Making the thickness of the plates |", and the diameter of the rivets f. 
18. Draw plan and section of the butt joint shown in Pigs. 398, 399. Full size. 

14. Set out a single-riveted lap joint for i" plates, making the rivets a suitable diameter, and pitching them as close together as you can, consistent with good 
practice. Full size. 

15. Draw three views of a J" hexagonal-headed bolt. You may make it long enough to hold together two plates whose total thickness is 3". 

16. Set out a pair of lock nuts for a IJ" bolt. 

17. Make full-size drawings of a 1" castle nut. 

18. Practise drawing different-sized bolts, using the table for dimensions, when required. 

19. Set out a double-riveted lap joint (chain riveting). Figs. 394, 395, making the thickness of the plates J", diameter of rivets }|", and the distance between 
rivet lines IJ". 



CHAPTER xxrn 



MISCELLANEOUS DRAWING EXERCISES IN WOODWORK, BRICKWORK, AND MASONRY 

226. Introduction. — In Figs. 369 to -387 you have some examples of joints, etc., in woodwork shown, which you, no doubt, have studied 
and drawn. The following pieces of woodwork may now be set out. You will notice that to avoid apparent complication no attempt 
has been made to show in detail the joints. After you have drawn these, or, for the matter of that, whilst you have them in hand, 
no doubt you will be carefully examining any such pieces of woodwork you may come across, and trying to understand how they 
are pieced together. 

227. Wooden Stand for a Machine. — The framed stand (Fig. 416) is typical of the kind of support largely used by machinists 
for light machines. The figure shows a pictorial sketch of a stand that has been measured for the purpose of setting it out. By 

this time you will experience no difficulty in making such a sketch from the actual thing and 
running your rule over it for the dimensions. The elevation and end elevation of the stand (Figs. 
417, 418) present no difficulty, so draw them to a scale of 3" = 1 ft. 

228. Kitchen Table. — The sketch (Fig. 419) shows the table upside down, for convenience of 





Pig. 416. — Isometric sketch of wooden 
stand for a machine. 



Fig. 419. — Pictorial sketch of kitchen table, 
inverted. 



Pigs. 417, 418. — Front and end elevations of stand. 

better showing the Avay the legs are arranged. Draw the two views (Figs. 420, 421) and add a plan, dotting in the legs, etc., to a 
scale of li" = 1 ft. 



132 



MISCELLANEOUS DRAWING EXERCISES IN WOODWORK, BRICKWORK, AND MASONRY 



133 



229. Diniag'-room Sideboard.^-This simple piece of furniture has been measured, as shown in the pictorial sketch, Fig. 422, and 
two views (Figs. 423, 424) have been drawn in orthographic projection. Set these out, scale 1^" = 1 It. 





Figs. 420, 421. — Front and end elevations of kitchen 
table. 



Fig. 422.— Pictorial sketch of sideboard. 



Figs. 423, 424.— Front and end elevations of 
sideboard. 



Brickwork and Masonry, 

230. You cannot walk very far, whether it be in the streets of London or in some large village, without coming across a 
bricklayer at work. You may have noticed that in building a wall he is very careful to place the bricks in accordance with some 













H 


C 








s 





^\GAl%.P/tw orsireti-/ii.'r Course 



?\&AZbPlon of Haider CoursP- 



S 

s 







s 














H 


C 









H 






J 


c 

Q 
4. 

I 






S 



fiG.^iPlimt^Seconjcl. Course 



fiGA26J>/an of StretcherCovrsc 



fioA-ZO-P/ofi of Header Course 



V\aA'iA.Plan.ofBottcm Course 



sT 



HI H?it^ 



311 



^7777777777777777^T7T777T, 



H H C H 



STRETCHER 



vTTTTTTTTTTTTTTTTTfTTTVTTTTTTTTTTryTTT 



■ 9"~ 



x: 



'>//>//^////^. 



I '! Hi kl 



H /3^"- 



W->/^' ■'TTTTTTTTTTTTT^ 



II 1 Hi 


II 1 1 s M 


1 1 




H 


s - 


1 1 4^. 


\ 


S 


|c| 


1 Y-^ 


M 


H 


[stretcher 









^'9"-^ 



•WJ//W/MJ, 



Figs. 427, 428.— Front and end elevations of 
9" wall. English bond. 



Figs. 431, 432. — Front and end elevations of 
14" vpall. English bond. 



Figs. 485, 436. — Front and end elevations of 
9" wall. Flemish bond. 



V, 



INDUSTRIAL DRAWING AND GEOMETRY 



set rules. It would take him a very long time to tell you about these. And if he did, you would probably not understand very 
much, or, at least, remember it if you did. However, there is no reason why you should not try to understand a few simple 
matters relating to this work, and this can be best done by making a drawing or two, which should tend to cultivate your powers 
of observation, and lead you to take a greater interest in such work. We will commence with — 

231. A 9" Brick Wall. — Look at Fig. 427, and then at Fig. 435. These elevations differ in the way in which the bricks are 
arranged in alternate courses or layers. The former is known as English bond, and the latter as Flemish bond. The term bond 
being given to any arrangement of bricks in which no vertical joint of one course is exactly over or above one in an adjacent 
course. In the English bond (Fig. 427) you will notice that the bottom course consists of bricks laid 
in the direction of their length, and therefore called stretchers, whilst in the courses H they are laid 
across the wall so that their ends are only seen in elevation. These ends are said to be headers. 
Now look at the plans of these courses, Figs. 426 and 425 (also the end elevation, Fig. 428). You 
will see that when one course is laid over the other, some of the joints cross one another, whilst others 
that are parallel are separated by a distance of half the breadth of a brick. Of course, to make this 
arrangement possible half bricks, C, called closers, have to be used near the ends of alternate courses, 
as shown. Comparing the English and Flemish bonds you will see that the former has more headers 



Bi;'5 



FiGAZlPfan. of Sfcond Course 




'TTTTTTTTTTTTTTT? 

Fig. 4-43. ELEVAxroN. 











C 










B 


S 


B 





FlG.438 Plan of Bottom. Gsurse 



/V/W-^X. 



I I " S IB) ^ 



-/3i"\~ - 



■777777^ '7Xr777y77777P> 



(II 




^ 




r 


1 1 




V 




1 


1 1 




1 1 1 1 












II II 












1 


1 I II 


^<- 




1 1 


1 ill! 


1 1 


II II 


1 1 








II II 




1 1 h 




11 1 


II II 








I 1 1 1 1 


1 1 






1 1 I 1 










--22i"-^ 


-//i". 


II II 




t 




1 1 1 




1 I I 


1 II 



















i 






\ V 

















Pigs. 439, 440.— Front and end elevations of 
14" wall. Flemish bond. 



Figs. 441, 442. — Front and end elevations of i 
quoins with bevelled edges. 



Fig. 444. — Plan of small church. 



for a given height and length of the wall than the latter, and is therefore better bound or held together, and forms a stronger 



MISCELLANEOUS DRAWING EXERCISES IN WOODWORK, BRICKWORK, AND MASONRY 



135 



structure ; for this reason civil engineers prefer this bond for their works ; whilst, ou the other hand, where appearance is an im- 
portant factor, as in architectural work, the Flemish bond is generally used. 

Make Drawings of the two 9" walls as shown, scale 1^" to the foot. Note the average size of our stock bricks is 
8j" X 4-f" X 2f " or, including one thickness of mortar, 9" x 4^" X 3". Thus a so-called 9" wall is really an 8|", and a so-called 
14" wall 131". 

232. A 14" Brick Wall. — After giving attention to the previous article, a careful inspection of Figs. 429 to 432, and 437 to 440 
will enable you to uuderstaud how the bi'icks must be arranged in the alternate courses to be properly bonded for the two different 
bonds in common use. You will notice that to secure the proper break of joint in the Flemish bond h bricks, B, called bats (Figs. 
437, 438), have to be used. 

Draw the views shown of both walls to a scale of 1.V' to the foot. 



KEYSTONE 





-2 0-' 





SKEW BACK 



Pigs. 446, 447. — Front elevation and section on CD of stone 
semicircular arch. 



Pig. 448. — Pictorial view 
of voussoir A. 



11//////// 
l,i/////// 
I //////// 

\ Will,/// 

\ \\lll,// 

Pig. 449. — Segmental arcli of a stone bridge. 



233. A Small Church in outline is shown in Figs. 443, 444. You will see that the steeple is represented by a square pyramid, P, 
resting on a square prism, S, whilst the top pieces of the other parts are equivalent triangular prisms, and these are supported by 
rectangular prisms. 

Make drawings of this structure as shown. Scale \" to 4 ft. 

Also an elevation looking in the direction of the arrow A. This view you should be now able to draw, with the exercise of a 
little ingenuity, and the assistance of Problem 192. 

234. Angle ftuoins with Bevelled Edges.— Look at Figs. 441, 442, you will see represented a combination of stone and brickwork 
you have often come across. The stone quoins are always made equal in thickness to a multiple of the thickness of the bricks. 



136 



INDUSTRIAL DRAWING AND GEOMETRY 



~7777777777^ TT 



'///////// 



'i-7i 



////////A//. 



YZM I \ 



'i-y&Ah^. Section ihroughline C D. 



y//////y//// 



c— 




Fig. 451.- 



-Elevation of square-headed window 
with flat-gauged arch. 



Pig. 452. — Section 
ot window sill, etc., 
on line AB. 



therefore in this case the thickness of quoin equal 9", due to 3 bricks, 
and the length and breadth of the quoins are usually multiples of 2^", the 
quarter of a brick's length. 

235. Stone Semicircular Arch. — The three views (Figs. 445 to 447) of 
this arch should now speak for themselves. The section (Fig. 447) is 
taken through the keystone, K, and the joints of the voussoirs on the 
intrados ^ projected from the elevation (Fig. 446). One of the voussoirs, B, 
is shown in oblique projection in Fig. 448. Draw the views shown to a 
scale of I" to the foot, and the voussoir B to a scale IJ" to the foot. 

236. Segmental Arch of a Stone Bridge. — The view shown (Fig. 449) 
is not a working drawing, but it gives you a good idea, particularly after 
working the previous exercise, of a type of structure that requires con- 
siderable skill to design on a large scale. 

237. Square-headed Window with Flat-gauged Arch. — If you use your 
eyes when you have a chance of going over a house in course of con- 
struction there are many parts, such as the window shown in Figs. 450 to 
452, that will be of interest. You will not be able to concentrate your 
attention on many of these until you have tried to make such drawings as 
those shown, and endeavoured to understand them. It should be explained 
that the dotted arch takes the load over the window opening, and that 
the joints shown on the gauged arch are not always the actual joints, but 
ones that are pointed on by the bricklayer in finishing the work. The 
top of the stone sill. Fig. 452, slopes slightly downwards to allow the water 
to run off, as you will see, and is therefore referred to as the weathering. 
To prevent the water flowing on to the brickwork below, the under edge 
of the sill is channelled, and this groove is called the throating. 

The figures are dimensioned, and no doubt you would like to make 
drawings of the window. A suitable scale would be i" to the foot. 



' The under surface ofithe arch, as shown in Pig. 449. 



CHAPTER XXIV 



VARIOUS MACHINE-DRAWING EXERCISES 

238. In Figs. 453, 454 you have a dimensioned sketch (elevation and end view) of a cast-iron bracket used to support a handrail. 
It is secured by a f" bolt. Make a working drawing of it, showing the two views given, and a plan. Also showing the bolt hole, 
which you are to put in a suitable position. Scale full 
size. Draw the circular parts first, and the plan before 
you decide upon the position of the hole. 

239. The Figs. 455, 456 show two dimensioned vievi^s 
of a cast-iron foundation washer : one form of the washers 
used in connection with the bolts that hold down heavy • y\ ' • 

machinery to concrete foundations, or with bolts that are A~'l 1 V^r'"^. B~ 

sometimes used to keep opposite walls of a building from 
bulging out. The washers distribute the pressure (due to 






Figs. 453, 454. — Dimensioned sketches of a cast-iron bracket. 



Pigs. 455, 456. — -Cast-iron foundation washer. 




Pig. 457. — Engine fitter's square. 



the bolts) over a large area. Draw the two views shown and add a section through the line AB, also a plan of the under side. 
Scale half size. Draw the circular view first. 

137 



138 



INDUSTRIAL DRAWING AND GEOMETRY 



240. A dimensioned sketch of an engine fitter's square is given in Fig. 457. Draw a plan and elevation of it. Scale 
full size. 

241. Two views of a gun-metal flanged bush are shown in Figs. 458, 458a. It is used to bush what is called a loose belt pulley, 
that is, a sort of idle pulley which is free to rotate upon a shaft, so that when the belt is not required to drive the fixed pulley 
(placed close to it, side by side), which is keyed to the shaft and rotates with it, it is made to run on the loose pulley ; so that the 
hole gets worn in time ; or, rather, the bush does. The latter is made a driving fit (as it is called) in the hub (or boss) of the wheel. 
That is to say, the fit is so tight that the bush is driven in by a hammer. When the bush has worn slack, the wheel can be 
rebushed, and made to run as true as ever. 

Draw the two views, but make Fig. 458a half elevation and half section through the axis. Scale full size. Commence on 
the end elevation, and remember that the diameter of the shaft on which it runs is 2^". 

242. Link End with Spherical Seat. — In Figs. 459, 459a are shown the elevation and section (on line AB) of a link end with 
spherical seating. You will notice that the spherical bush or seat is split, to enable you to get it into and out of position. If one 
half of the seating be moved round about the axis of the rod a quarter of a turn, it can be withdrawn, after which tlie other part 
can be manipulated in the same way. With such an arrangement the link or rod can have a small movement about its axis with- 
out the pin (which fits the hole) binding in the hole. , 
Draw the two views, full size, commencing on the circles ''^ 
in the elevation. 

243. Joint Pins.— In Figs. 460 to 462a three pins _ ^^^^^^ 
for Tpm joints are shown, with different methods of / ^""1"^^- ' j ■ ^^^^^^^' "T 




w;mm!^^y^^mm>^Mf;^ 



" to 



\^mm;^M4^^^>^^y;y(^y 





I 



Figs. 458, 458a.— Gun-metal bush. 



Figs. 459, 459a. — Front elevation and section on line AB, of a link end with spherical seated bush. 



securing the collar. In 460, 460a a taper pin passes through the pin end and is seated in the groove in the collar. In Figs. 461, 
4.61a a split pin (a detail of which is shown in Fig. 463) passes through the pin end, and its tail is spread open, as shown, to prevent 
it coming out. Figs. 462, 462a show a taper pin passing through the collar ; the hole is made taper by using a taper reamer 



VARIOUS MACHINE-DRAWING EXERCISES 



139 



after the hole is drilled. These taper pins usually have a taper of ^" to the foot (or are made to Morse taper), so that if they are 
driven home they hold tight by the wedge action ; but when they are likely to be subjected to much vibration they are made with 
a split end, being either forged with a split, or sawn, as in Figs. 464, 464a. The proportions shown are those in common use. 
Exercise. — Draw the three pins full size, and the details twice full size. 



JOINT PINS. 





-H3i<- 



COU.AR I I . 
WAftHER. r 'a 



Pigs. 460, 460a.— 
Arranged with 
taper pin. 



Figs. 461, 461a.— 
Arranged with 
split pin. 




Pigs. 462, 462a.— 
Arranged with 
taper pin through 
collar. 



CIRCLE GIVES POSITION/ 


1 >v 


OF HEAD / 


' jh\ 




^-^^-tw 


1/ v^ 


•Jl*/ X^^J -1 " 






a--£ii^^^ -1* 


K V^ 


V ^^ i ^\ / 




' V/ 






' \ / / ' 




1 >Sfc / 




J^-^^oV 




'J. / 




' / / 




/ ^ 

; / 



Pig. 463.— Detail of split pin. 



J" 



STEtL PIN. 



S2 SAW CUT. 



-M 



-IF 



Pigs. 464, 464a.— Detail of taper split pin. 



244. Belt Palley. — Figs. 465, 466 are two views of a heavy pulley for a double 3" belt, the arms are segmental in section, as in 
Fig. 467, or elliptical, as in Fig. 468. Draw to scale of half full size. Note that the breadths of the arms are set off at the centre 
of the hub or boss, and at the rim. Commence with the circular view. These drawings should present no difficulty now. 

245. Locomotive Crank. — A very simple form of locomotive crank, from a drawing point of view, is shown in Figs. 469, 470, 
It is a pattern introduced Ijy Mr. Wordsell, the famous locomotive engineer, some years ago. To make a drawing of it, use a scale 
of l.V to the foot. Commence on the end view. Fig. 469. 



140 



INDUSTRIAL DRAWING AND GEOMETRY 



246. Cast-iron Bearing Block. — A very simple and inexpensive form of bearing is shown in Figs. 471, 472. The drawings will 
speak for themselves now. You should draw this piece full size, and commence on the sectional elevation. Also draw a plan. Be 

careful to find the proper position of the centre C, so that the lines 
flow into one another. A pictorial view of the block is shown in 
Fig. 384. 

247. Gun-metal Tail-guide for Valve Rod. — Figs. 473, 474 show 
two views of a guide you may have noticed projecting from the 
end of the slide-valve jacket of a steam engine ; its function is to 
steady and keep in its place the end of the valve rod, so that the 
valve may not leave the face upon which it works. In making 
the drawings (scale full size), commence on the end view, Fig. 
473, and project from that the elevation, from which project a 
section on the line CD. You will have to be very careful in doing 






Pigs. 465,466. — Elevation and sectional elevation of a straight arm belt-pulley. Pig. 467.- 




PiGS. 469, 470. — Locomotive crank, WordEell's pattern. 



Section of segmental arm. Pig. 468. — Section of elliptical arm. 

this to think out which lines will be seen- 
after the body has been cut along the section 
line. By now you should be able to do this 
very well. 

248. Couplings. — Whenever a line of 
shafting exceeds some 20' in length it is 
made up of two or more lengths, connected 
together by what are technically called coup- 
lings, many forms of which are in use. One 
of the simplest of these is the butt-muff 
coupling, three views of which are shown in 
Figs. 475, 476, and 477. They are arranged 



VARIOUS MACHINE-DRAWING EXERCISES 



141 




Figs. 47], 472. — End elevation and sectional elevation of oast-iron bearing block. 



to form a drawing exercise, in continuation of the previous ones, and it will be convenient to touch upon the principal features of 

the arrangement as we describe how it may be drawn. 

249. Drawing Exercise. — To draw a Butt-Muff Coupling'. Scale half full size. — From an inspection of the figures ^ it will be 

seen that the sleeve, muff, or box, B, is slid over the ends M and N of the two pieces of shafting that butt, and are required to be 

coupled together, and a taper key, K, is used, as 

shown, to fix the box to the shafting so that one 

length may transmit a torque, or twisting action, 

to the other. jSTow, remembering what we have 

said about commencing a drawing of an object that 

has a circular part, it will be seen that this is a 

case where the end views. Figs. 475 and 477 (or as 

much of them as possible), should be drawn first ; 

so, having drawn the circles, the section of the key 

(taken through the centre of the coupling). Fig. 475, 

can be set out. As this is an important detail, it 

is shown in Fig. 478 to a larger scale. The point 

A, on the centre line and circle, is the centre of the 

section, and the thickness of the key here should be 

half its breadth. Now, the rule for breadth W may 

d 'ih 

2 -f -J", then in this case, W = -^ 

BC = I", so the depth, AC, of the keyway (which is uniform 
throughout the length) '^ becomes £', the full taper of ]^" to the 
foot being given to the keyway in the box. Fig. 479 shows 
the key in pictorial projection. With these hints, you should 
now experience no difficulty in drawing the three views shown, 
and in setting out a complete plan of the coupling. You will 
notice that you are instructed to make the drawings to a scale 
of half full size, that is to say, you are to draw the object one- 
half its real size, but you will not dimension the drawing with 
figures one-half of the original ones, as tlie dimensions on a 
drawing indicate the real size, and are independent of the scale 
to which the drawing may be made. All horizontal dimensions 

' It will be seen that the proportional parts in terms of the unit used (d + i") in designing it are given, but it has been also dimensioned for a 3 
drawing exercise, and you only need pay attention to the actual dimensions given. 

- The taper is always made on the coupling or boss, which is fitted to the shaft, excepting when the key is flsed, and the boss moves along the shaft a short 
distance ; the key (which is then called a feather) is then parallel. 



be, ^ + -J", then in this case, W = ^ 4- I" = 1", and therefore 




Pig. 473, 474. — End elevation and elevation of a gun-metal tail guide for valve rod. 

shaft as a 



142 



INDUSTRIAL DRAWING AND GEOMETRY 



are placed to read borizontally in the spaces left for them between the diinensLoa lines, and all vertical dimensions read from 
bottom to top of drawing when looking from its right-hand edge. The points of the arrow-heads must touch the lines between 
which the dimension is taken. 

Every important part should be dimensioned on at least one of the views, and in cases where a body consists of two or 



BUTT-MUFF COUPLING. 






Fig. 475. — Section on line e/. 



Fig. 476. — Sectional elevation. 



Fig. 477. — End elevation. 



more divisions of its length, breadth, or thickness, the overall (sum of its parts) dimensions should be shown; indeed, in 
some cases it saves time in reading a drawing (when it gets into the works) if important dimensions are occasionally repeated 
on different views. 



^ w=af + 




UNIT = c/ + ^' 



; in 



SKETCH OF KEY 



PosirroN OF sunk kc 
SECTION eF 



2 

TAPER OF KEY 
INCH PER FOOT 
OF LENGTH 




HEAD. 



POINT. 



Fig. 478. — Proportions and position of sunk key. 



Fig. 479.— Sketch of key. 



It should be explained that although the muff-coupling is the simplest one in general use, it requires to be very carefully 
fitted if it is to be a first-rate job, for, obviously, unless the depth of the keyway in each of the shafts to be coupled be exactly the 
same and the diameters be the same, the key will be bedded on one shaft whilst the other will be loose. To prevent this 
happening, some engineers make the key in two lengths, and drive them both in from the same end, one for each shaft. Or they 



VARIOUS MACHINE-DRAWING EXERCISES 



143 



may iDe driven from opposite ends, as shown in Fig. 480. This fignre and Fig. 481 also show how the coupling is cased to protect 
the clothes of workers from coming into contact with the key-heads. 

Proportions for other Sizes. — Taking the unit as d + ^", the usual proportions are shown on the figures in terms of the unit 
for other sizes of the shaft. As a further exercise, you might make drawings of such a coupling for a 2" shaft. Full size. 

Materials. — The box is made of cast iron ; the shafts, usually of mild steel or wrought iron ; and keys, of mild steel. 

249a. Drum or Barrel of Hoisting Machine. — A section. Fig. 482, and end elevation, Fig. 483, of a drum or barrel of a hoistin<' 
machine, such as a crane or crab, are shown. The axle is supported by bearings at its journals J, and the spur drivint^ wheel 
is fixed on the end A. It is fully dimensioned, and the two views can be drawn half full size. Note. — The keys which secure the 
barrel and wheel to the shaft are shown on the end view. 

16 ':>ixs 





^-^■ ^ I 




N 



-Tit-— /J"-— - 



"il^J. I 



FiGS. 480, 481. — Sectional elevation and end elevation of cased 
butt coupling. 



\^^',■.'.','-^■^'-'.^w-^'-^■■-■~'■■~^k^||>.'-w^^^^^^'^^' 



Pig. 482. — Section of drum or barrel of hoisting 
machine. 




Fig. 483.— End elevation. 



250. How to measure the Diameter of a Large Cylindrical Body. — If you had a pair of calipers large enough, the diameter of a body 
such as the rough drum, Figs. 482, 48.3, could be accurately measured, but your small pocket calipers would not be large enough for 
the purpose. However, you may take a piece of fine string, or, better still a tape, and measure the circumference of the drum, from 
which you will be able to find the diameter thus : The circumference in this case measures 27'48", but diam. x ^ = circumfer- 



ence very nearly, or the diam. = 



7 X circum. 7 X 27-48 



= 8-75". 



An advantage of this method is that the measurement is truer 



22 22 

when the body is not quite round, than if it had been made with the calipers, as it gives the mean diameter. 

251. A section and the plan of the under part of a petrol engine piston are shown in Figs. 484, 485. It is of the ordinary trunk 
type, fitted with three rings, ff" X I" section made of good grade cast iron. The steel gudgeon-pin is held in position by two ij" set- 
screws. Draw the views to a scale of full size, commencing with the plan. Such pistons for high-speed engines are made lighter. 

252. Petrol Engine Connecting Eod. — The views of the connecting rod (for the piston in Art. 251) shown in Figs. 486, 487 are 
fully dimensioned, and should now speak for themselves. Make separate drawings of each part of the rod. Scale full size. 

253. Cast-iron Bracket with Pin.— The pictorial views ^ (Figs. 488, 489) of the bracket and pin are fully dimensioned, 
and you may draw an elevation looking on the face AA, a plan, and a sectional elevation taken through the axis BB, projected from 
the first view. Scale full size. 

' Talien from the 1909 Board of Education paper in Machine Construction and Drawing, by kind permission of H.M. Sationery Office. 



144 



INDUSTRIAL DRAWING AND GEOMETRY 








Figs. 484, 485.— Petrol engine piston. 



Figs. 486, 487.— Petrol engine 
connecting rod. 



Pig. 488. — Front elevation of oast- 
iron bracket with pin. 



Fig. 489.- 



-Back elevation 
bracket. 



of 



CHAPTER XXV 

INTERSECTIONS AND DEVELOPMENTS OF SIMPLE SOLIDS 



Intersections. 

254. Intersections. — The determination of the lines of interpenetration due to the intersection of two solids is a branch of advanced 
geometry beyond the scope of this work, but there are a few very simple but important cases which you will understand. Some of 
these are based upon the fact that if two cylinders, or two cones, or a cone and cylinder, envelop a common sphere (that is to say, 
both solids enclose the same sphere), then the elliptical joint or section on the one enveloping body is exactly the same in shape 
and size as that on the other, and the projection of the section or joint on a plane parallel to the axes is a straight line. The follow- 
ing examples will make this clear. 

255. Intersection of Two Equal Cylinders. — Fig. 490 shows two equal cylinders with axes ac and he intersecting in c (the centre 
of the sphere), to form an elbow ; the line de being the elevation of the line of intersection ; the true shape of the intersection, of 
course, being an ellipse. Another case, that of two equal 
cylinders, is shown in Figs. 494, 495 ; the two parts T and E 
forming a tee piece, as it is called. 

256. Intersection of CyUnder and Cone. — The two axes 
(Figs. 491, 492) intersect in c, the centre of the enveloped 
sphere, and the intersection of the surfaces at the sides gives 
de, the section (in each case), whose true shape you will know 
by this time must be an ellipse. The two parts of Fig. 491 
may form a conical ventilator, and those in Fig. 492 a conical 
nozzle. 

257. Intersection of Two Cones and a Cylinder. — The three 
intersecting pieces (Fig. 493), if in the form of pipes, would 
niake what is called an irregular breeches-piece. It will be 
seen that they all envelop the sphere whose centre is c ; the 
intersection of the cylinder and cone cv is obviously on the line ef, and of the two cones on the line dg ; these lines intersect in 
i, so join this point to k, and the intersections id, ie, and ik, of the three solids are complete. 

U5 h 




r- 




Pig. 490. — Intersection 
of two equal cylinders. 



Pig. 491. — Intersection of 
cylinder and cone. 



Fig. 492. — Intersection 
of cylinder and cone. 
Second case. 



146 



INDUSTRIAL DRAWING AND GEOMETRY 



258. Intersection of Two Unequal Cylinders. — A very interesting and important case is shown in Figs. 494, 495, where we have 
the axes of the two unequal cylinders R and S not in the same plane. The points ee' and ff in the intersection can be at once 

found, as shown, and by using an auxiliary elevation any additional 
number of points can be found, such as dd" ; dividing the semicircle of 
the auxiliary elevation into a suitable number of parts (in this case 6), 
and measuring the distances, such as b'd', and marking them off above 
and below the axis m'm', as at b"d". The Figures should now speak 
for themselves. 

259. Intersection of a Circular Fillet and Plane Surface. — Look well 
at the tee-end of the connecting rod in Figs. 496, 497. You will see 
that the intersection of the fillet and the sides of the tee-head is the 
curve b'c'd'c'. The question arises how this curve is to be properly 
projected so as to truly represent the edge or line of intersection on 
the actual piece. A careful examination of Figs. 496, 497, should 
make this clear. The plan of the highest point in the curve will be b, 
and the elevation of the circular section containing it gives the position 
b'. Divide b'h into, say, three parts, and draw the lines 2 and 3 
parallel to op, then these will give the diameters of the circles 2 and 3 
in the plan, which circles cut the sides of the head in the points c 
and d, projectors from these points giving the points c d' in the required 
curve in elevation. The point e', the lowest one in the curve, is found 
by drawing a projector through the centre a' of the fillet, to a, in the 
plan and with centre n, radius 7ia, sti iking the arc eae, then a pro- 
jector from e cuts the line op in the elevation in the required point e. 
The points on the left side of the curve are found in the same way, or by symmetry. 

260. Intersection of Circular Fillet with Cylindrical Surface. — Look at your bicycle cranks, and notice where the arm merges into 
the boss or hub there is a curve, something like the one b'e (Figs. 498, 499) in the elevation. Obviously the arc bed in plan is on 
a cylindrical surface which intersects the circular fillet between the arm and boss. To find b', the highest point in the curve, 
join the centres m and n (in plan), cutting the circle in b, through which drop a projector, lb', cutting a horizontal through a (in 
elevation), the centre of the fillet arc, in b'. The other points are found as shown in the figures, and explained in the previous case. 




Pig. 493. — Intersection of 
two cones and a cylinder. 



Pigs. 494, 495. — Intersection of cylinders. 
Two cases. 



Developments. 

261. Introduction. — Take a model of a cylinder, and cut a piece of paper so that when it is wound round the body of the solid it 
exactly fits the cylindrical surface ; the shape of this sheet will be a rectangle whose breadth equals the length of the axis of the cylinder, 
and whose length equals the circumference of its base. Now, this sheet of paper may be referred to as the development or (lay-out) 



INTERSECTIONS AND DEVELOPMENTS OF SIMPLE SOLIDS 



14.7 



of the cylindrical surface. And of coarse yoa could place the sheet on the drawing board and roll the cylinder over it, keeping 
it in contact with the paper throughout the whole of its length during a complete revolution. 



c ei 





Figs. 496, 497. — Intersection of circular fillet and plane surface. 



Fios. 498, 499. — Intersection of circular fillet with cylindrical surface. 



262. Development of the Five Regular Solids. — You can now understand in what sense Pig. 500 is referred to as the develop- 
Dient of a tetrahedron (refer to Fig. 307). You will see that if you set out this figure, consisting of four equilateral triangles, each 
representing a face of the solid, you could fold it into a model of the solid. Do this with some stiff paper or millboard, and fasten 
the edges where they meet with sealing wax, or leave a little margin on the edges (as shown dotted), and gum or glue them together. 
You will now better see that Fig. 501 is the development of a cube; Fig. 502 that of an octahedron (refer to Fig. 591), consisting 
of eight equilateral triangles ; Fig. 503 the development of a dodecahedron (Fig. 592), consisting of twelve pentagons; whilst 504 is 
the development of the icosahedron (refer to Fig. 593), consisting of twenty eqviilateral triangles. 

If you are fond of such work, and have patience, you might make models of these interesting solids, and of others. 

263. Development of a Square Elbow. — When two equal pipes, with ends bevelled at 45°, are brought together (as shown in 
Fig. 490), they form what is called a square elbow. Now, suppose you wish to make one out of metal plate, you will have to cut each 
plate to the exact shape, so that when it is bent round into the form of a cylinder one end will be bevelled, as in Fig. 505 ; in 



148 



INDUSTRIAL DRAWING AND GEOMETRY 



other words, you would want to find its developinetit; and Figs. 505, 506 sho\v liow this is done. Divide the semicircle in Fig. 505 
into, say, six equal parts, and draw the lines shown from these divisions, making the base line dd, in Fig. 506, n-D in length, or 
^D (where D is the diameter of the pipe) ; or you may step off the divisions along dd, making them equal to those in the semi- 



/^ JOINTING 




Fig. 500.— Develop- 
ment of tetrahedron. 





Fig. 501.— Development of cube. 



Fig. 502. — Development of octa- 
hedron. 



Fig. 503. — Development of dodecahedron. 



circle ; but, of course, that is not so exact. Number the divisions, as shown, and draw the vertical lines from the divisions to 
intersect horizontal lines from the points in the joint line cd (Fig. 505) in points in the required curve dcd (Fig. 506), which can 
be neatly drawn freehand through the intersecting points. Or this development may be made by the following direct method. 




Pig. 504. — Development of icosahedron. 



c 


















i 


5 




<\>. ^-^ 










\ 


-H /I 




N 










<^> 


/ 










\ 








^" ^ 














s. 






\ 


i f -^ 














s^ 






















Kyi \> 





1 2 3 4-5 654-32 10 

DEVELOPMENT 
^ 22 rx 








\j 






-^ 


7" 

GIRTH LINE 






0/234-5 65432 / 
DEVELOPMENT 



22 



D 



j£. 



GIRTH LINE 



Figs. 505, 506. — Development of square elbovr. 



Fig. 507. — Development of square elbow. 
Direct method. 



264. Direct Method. — Fig. 507 shows a direct method of setting out the curve, which should speak for itself when compared 
with the other figures. 

265. Development of a Square Tee Piece. — Fig. 495 shows the axes ola and cc of two cylinders^ which form a square tee piece ; 



INTERSECTIONS AND DEVELOPiMENTS OF SIMPLE SOLIDS 



149 




Fig. 508 shows the end view of such a piece. To draw the development, Fig. 510, first draw the semicircle in Fig. 508, and divide 

it, as shown ; through the three divisions erect perpendiculai's cutting the upper circle in 1', 2', 3', 0' ; horizontals through these 

points cut the ordinates from the base or girth 

line (Fig. 510), and give points in the required 

curved boundary line. The development of the 

hole in the pipe mn is shown in Fig. 509 ; it is 

symmetrical about the line aiai,and thedistance 

3iO is found by stepping off distances 3i2i and 

2ili, also liO equal to 3'2' and 21', also I'O in 

Fig. 508. For the widths : reading the two 

views together, «i3i is eqvial to a3 ; 5i2i equal 

to 62, and Cili equal to cl. 

266. Development of a Cone. — If the plan 
and elevation be drawn as in Fig. 511, and 
each semicircle be divided into, say, six equal 
parts, then the arc 0.6.0 can be drawn with 
V, the vertex, as centre, and L, the length of 
the slant side, as radius ; take the length of 
the chord 01 (one of the divisions) in the 
semicircles as an opening of your compasses, 
and step off twelve divisions, along the arc as 
shown, to determine its length, then join the 
last one to V, and the sector S is the required 
development. Note : the sector is only the 
true development when the length of its arc is 
equal to the circumference of the cone's base. 
Obviously the equal chords are measured on 
circles of different curvature, so the lengths 

true development, make the angle Q = — ^ 



Pig. 508. — Square tee 
piece. 



Fig. 510. — Development of pipe 
for square tee piece. 





Pig. 509. — Development of the 
hole in pipe nin. 



Pig. 511. — Development of cone. 



of the arcs they represent cannot be quite the same. Therefore, to get the 

Because it can be proved that 6 : 360° : : E : L, where E is the radius of the cone's 

base. In Fig. 511 L = 3E, .-. 6 = ^^ = 120°. 

267. Developments of Pyramids. — If you have read Problem 191, and examined or drawn Fig. 315, you will see at once that 
Acd in Fig. 512 is the true shape of a side of the triangular pyramid, and, therefore, a part of its development; the base cdb is also in 
true shape, and it forms another part. It only remains to deal with the other two sides, and this is conveniently done by taking 
A as centre, radius Ad, and describing the arc cdci. It only then remains to mark off the points bi andci wijth an opening of the 
dividers equal to the edge cd. Join Aci and A6i ; then the figure cAcibidb is the development of the solid. And on the same lines 



150 



INDUSTRIAL DRAWING AND GEOMETRY 



you will now be able to treat a square pyramid as in Fig. 513, or a frustum of it, as in Fig. 514. To complete the development in 
Fig. 514 you must find the true shape of the section of the pyramid made by the cutting plane. This you can do as in Fig. 312, 
Problem IS'.K But the construction shown in Fig. 514 should now be easily followed. 




Fig. 512. — Development of a 
triangular pyramid. 



PiQ. 513. — Development of a square Fig. 514. — Development of the frustum 
pyramid. of a square pyramid. 



EXERCISES. 

1. A square elbow is formed by two 3" pipes intersecting, as in Fig. 490. Draw the arrangement to a scale of half-size. 

2. A conical ventilator is formed by a 3" pipe intersecting a sheet-metal cone, as in Fig. 491. Draw the ventOator, making the angle of the conical part 60° 
at its vertex, and the diameter of the mouth part 8". Scale half-size. 

3. A conical nozzle is formed by a 4" pipe and a conical part, as in Fig. 492. Draw the nozzle, making the angle at the vertex of the mouth-piece 40°, 
and the diameter of the nozzle 2". Scale half-size. 

4. A tee-piece is formed by two 3" pipes intersecting at right angles, as in Figs. 494, 495 (the left-hand part). Draw two views of the arrangement. Scale 
half -size. 

5. A tee-piece is formed by the intersection of a 8" pipe and a 4" one, their axes being in the same plane and at right angles. Draw two views of the 
arrangement, showing the line of intersection. Scale half-size. Note. — The right-hand part of Figs. 494, 495 shows how the line of intersection is obtained. 

6. Figs. 496, 497 show the curve formed by the intersection of the tee-end and fillet of part of an engine connecting rod. Assuming that the diameter of the 
rod is 2", the radius of the fillet 1", and the breadth of the tee-head is 2", draw the curve of intersection. Scale fuU size. 

7. Figs. 498, 499 show the curve formed on a crank boss due to the intersection of the fiUet with the curved part of the arm. Assuming that the boss has a 
diameter of 2i", that the radius ra6 is If", and the radius of the fillet j", draw the curve as shown. Scale full size. 

8. Draw on stiff paper the developments of the following regular solids, cut them out, and fold them to form models of the solids ; if you make them with 



INTERSECTIONS AND DEVELOPMENTS OF SIMPLE SOLIDS 151 

jointings, as in Pig. 500, you can gum the edges together where they meet, (a) A tetrahedron of 2" edge. (6) A cube of 2" edge, (c) An octahedron of 2" edge. 
((f) An icosahedron of 1\" edge. 

9. Draw the development of an li" diameter square elbow, as in Figs. 505, 506. Scale full size. 

10. Draw the development of a square tee-piece, diameter lA". Scale full size. (Refer to Pigs. 508-510.) 

11. A cone has a 2" base and 3" axis. Draw its development. 

12. The cone in the previous exercise is cut by a plane bisecting its axis and inclined 30" to its base. Draw the development of its lower part or frustum. 

13. Draw an equilateral triangular pyramid of 2" edge of base and 3" axis, and show its development. 

14. Draw a square pyramid, edge of base 2", axis 3", and show its development. 

15. The square pyramid in the previous exercise is cut by a plane bisecting its axis and inclined 80° to its base. Draw the development of its frustum. 
Note. — If the developments in Exercises 11 to 15 be drawn with jointing, as in Pig. 500, models of the solids can be made by cutting out the figures, folding 

and gumming the edges. 



CHAPTER XXVI 

PRINTING, SHADING, TRACING, ETC. 

268. Printing, etc. — The following style of lettering is most suitable for notes or remarks on a drawing. The letters, etc., should be neatly written 
with a fine pointed writing pen of the ordinary type : probably the best for this kind of work is Perry's No. 120 EP. 



ahcd efg hijklmnopqrstuvwxyz 



In drawing office practice it is usual to stencil ' headings and titles, etc., in plain letters, such as the following, the size varying from J" to |", 
according to the size of the drawing ; for example, the heading or title on medium or royal size sheets would be in good proportion if made with \" 
letters for half imperial sheets, 22" x 15", |" for imperial, 30" X 22", with h" or |" letters for double elephant, 40" x 27", and |" or j" for antiquarian, 
53" X 31" ; and such sub-titles as plan, elevation, etc., with ^" or ^" letters : — 

ABCDEFGHIJKLMNOPQRSTUVWXYZ 

1234567890 

Although most of this printing is done by stencilling,^ you should endeavour by practice to do it neatly by freehand, to enable you to do finished 
work, or proceed when stencil plates are not available. The quality of printing and writing upon a drawing greatly adds to or detracts from its appear- 
ance. 

269. "Working Drawings of machinery are made in such a way that the form and size of every detaU are clearly shown for the guidance of those 
in the works. The rule is to make them to as large a scale as possible, generally full size for all small details, and i and j full size for larger ones. 
Such drawings are first carefully and completely set out in pencU,^ and then inked in if required. All parts out by section planes being hatched with 
sectional lines indicating the materials they are made of, in accordance with the shading shown in Fig. 524, or alternately, they are coloured * to indicate 
the materials, as explained in Art. 17. The edges of surfaces that are to be machined are usually coloured with a narrow band of a deeper tint, 

' A good deal of practice is necessary to enable the beginner to do this neatly. He usually commences by making the stencil brush too wet, which causes 
the ink to flow between the stencil plate and paper. The best expedient is to recess a piece of Indian ink in a thin block of wood, and, after wetting the brush, 
rub it over the ink and wood till it is dry enough to use on the plate. 

- It is best to start from the middle of a title when stencilling, so as to get it quite symmetrical with the drawing. This can easily be done by counting the 
letters which come each side of the centre, allowing one for each interval between two words. 

^ Many students and draughtsmen leave a large proportion of the details in their minds for inking, instead of completely pencilling them on the drawing. 
Nothing can be said to defend this objectionable habit, which puts an unnecessary tax on the mind. It has the forther disadvantage that the pencil-drawing 
cannot be passed on to another for inking in. To make a finished pencil-drawing quickly, neatly, and properly, you must patiently practise ; for it represents an 
art, and is seldom a gift. 

' Drawings from which tracings are to be made for reproduction by photographio printing to give blue prints, are of course always section-lined, and not coloured, 

152 



PRINTING, SHADING, TRACING, ETC. 153 

or hatched. The next step is to ink in with red ink ^ the centre lines,^ and the dimension lines with Prussian blue.-' The arrowheads and the 
dimensions should be now neatly written with a writing or mapping- pen of the kind explained above, care being taken to make the dimensions bold and 
neat, so that they can be easUy read from the drawing. The value of a drawing for workshop purposes greatly depends upon the clearness and accuracy 
of the figures or dimensions and the skilful way in which they have been arranged. Often an occasional duplication of a dimension on diilerent views 
will save much time in the works. 

In cases where orig-iual drawings are not likely to be much, used, it is the practice of many engineers not to ink them in. This, of course, necessitates 
more careful finisliing in pencil. Indeed, the beginner should not be encouraged to do any inking-in work until he has become fairly proficient in the 
somewhat difficult art of making a good pencil drawing. 

The particulars as to the scale to which the drawing is made must always be clearly shown upon the drawing, not in order to enable workmen 
to " scale it," as sufficient dimensions should always be given to entirely obviate this. If a di-awing is not completely dimensioned and there is any 
probability of it being sent abroad where a diiierent system of measurement is used, or to where it will be exposed to variations of temperature, the scale 
should always be drawn upon the drawing. Sometimes the scale of a working drawing has to be reduced to make it suitable for attachment to a speci- 
fication, or some such purpose ; in such a case, proportional compasses may be advantageously employed ; the best practice being to locate the centres 
of the circles and curves and to ink the latter in direct, and then to proceed with the straight lines, avoiding the use of pencils as much as possible. 

Shade Lines and Line Shading. 

270. Shade Lines. — The appearance of finistied drawings (which are usually made to a small scale) is improved, and the true form of parts made 
more inteUigible in a single view, by the use of shade or dark lines, which give an appearance of relief, to the various parts. 

Shade lines indicate the intersection of two surfaces, one of which is in the shade and the other illuminated. In arranging the shade Hnes, the 
parallel rays of light are conventionally assumed to come from the left and from behind (over the left shoulder) towards the object, their plans and 
elevations making angles of 45° with the vertical and horizontal planes respectively, their real inclination to the ground being 35°'15 nearly.* Thus, 
applying these rules to the body shown in Fig. 515, we have the back and right-hand edges ah and he, also e/and/^ of the projecting piece of the plan 
as shade lines ; whilst the rules applied to the elevation give us the bottom and right-hand edges, hi and ic' , as shade lines. But, it should be explained, 
the line hi would not be a shade line if the body was actually resting on a horizontal surface, as the two surfaces would be in contact, and the upper not 
projecting beyond the lower. For these reasons jTc is not a shade line but g'h is. These rules applied to a case where there is a recess or hole, as in 
Fig. 516, give us the front and left-hand edges, 6c and ah, as shade lines ; the upper surface being in the light or iUtuninated, and the front and left-hand 
sides of the hole in the shade. 

In dealing with curved surfaces shade lines are never used to denote their contour or outlines. Thus, in Fig. 517 the only shade line on the 
elevation of the vertical cylinder is de, the line representing the soKd's base,/e, being a boundary line of a curved surface, is not a shade Une. 

Now, the plan of the cylinder has a curved outline, and the rule relating to such cases is to make the shade line begin at the points a and 6, at 
which the projections of the rays touch this outline, and let it gradually increase in thickness till its full strength is reached at c. Similarly, for the hole, 
the shade line increases in thickness from m and n to g. 

The rules we have given relating to the rays of light we shall see are also concerned in the art of Shading ; but, strangely enough, although generally 
followed by artists, many English engineers prefer to take the rays of Kght as shown in Fig. 523, where the rays in plan are parallel to those in elevation . 
This makes no difference to the elevation, but in plan the shade lines come in front, as shown, instead of at the back. 

271. Shading by Lines. — By shading a projection of an object its true form can often be rendered intelligible in a single view. For example, the 
shaded view of a cylinder explains itself. But, on account of the time and labour involved, shading by tinting is only rarely used, even in finished 

' This may be prepared by rubbing down a little colour from the cake of crimson lake. The practice in some offices is to use blue ink for centre lines, and red 
for dimension lines. 

- The dimension lines on tracings prepared for blue prints may be drawn in red ink, and centre lines in Indian ink ; the latter are formed by alternating 
dashes (of I" and |" lengths) not too close together and evenly spaced. Only short dashes are used on short centre lines. 

' This may also be made by rubbing down a cake of the colour required, but most draughtsmen have the use of bottles of specially prepared red and blue inks. 

* The cosine of the angle being obviously the V2 -7- VS. 



154 



INDUSTRIAL DRAWING AND GEOMETRY 
EXAMPLES OF SHADE LINES AND LINE SHADING. 




Pig. C21. 



Tig. 523. 



PRINTING, SHADING, TRACING, ETC. 155 

machine drawings. However, a similar effect can be easily produced by a few shading lines/ which are or should be drawn in accordance with the 
rules followed in shading proper. To commence with a simple example that very often in many forms appears on machine drawings, we have in Fig. 518 
a vertical hexagonal prism, with its front face in the Hght or illuminated. Such surfaces parallel to the vertical plane would receive flat tints, 
and the nearer the surface is to the eye the lighter such tints would be, and the shading Unes would be equally spaced (between b' and c'), the 
spacing, beiug increased on the lighter sui'faces parallel to the plane of ijrojection, and the surfaces in the shade also receive flat tints, but the nearer such 
surfaces are to the eye the darker such tints are, or the closer the shade lines. Thus — 

Surfaces inlthe light inclined to the plane of projection have given them graduated tints (represented by graduated lines, as shown between a'. V , 
Fig. 518). and as such surfaces recede from the eye the tints are made darker, or the lines closer together, as shown. 

Surfaces in the shade inclined to the plane of projection, also have given them graduated tints (or lines), and as such surfaces recede from the eye 
they are made lighter, or the lines further apart, as between c and d' , Fig. 518. 

When two such surfaces are unequally inclined, the one upon which the rays impinge most directly is made lightest. 

Curved Surfaces. —The above rules in the main are followed in shading curved surfaces. Thus in Fig. 519 we have the plan and elevation of 
a vertical cylinder upon which the light falls from a' a" to e'e" , but most directly at the generator whose plan is 6 ; this, therefore, as we have seen, should 
be the lightest part, but succeeding generators from 6'6" to d'd" approach the eye, and according to what we have seen should therefore be increasingly 
lighter. So, in order to meet both these considerations, it is the pi-actice to bisect hd in c, and make the surface between h'h" and c'c" the lightest ; in 
fact, it is usually untinted, and remains white. Obviously, the darkest part of the cylinder is at e'e" , so that the shade and shading increase in depth 
from c'c" to e'e" , and diminish from e'e" toff. 

A horizontal cylinder, with its axis perpendicular to the vertical plane is shown in Fig. 520, and the student will see that similar lines are used 
in arranging the shading. The case of a vertical hollow semi-cylinder is sho^vn in Fig. 52] , and, for reasons we have explained, the lightest part of the 
cylindrical sxu-face is between the generators b'b" and c'c", and the darkest at the generator e'e" ; the part between e'e" and ff" being in the shade. 
Fig. 522 is a hollow horizontal semi-cylinder whose axis is parallel to the vertical plane, and the shading shown should now speak for itself. 

Note. — Pigs. 515 to 523 are first-angle projections. If they had been third-angle projections, the explanations given would equally apply. 

272. Copying Workshop Drawings. — The original drawings are kejit in the drawing office for reference piu-poses, and copies only, produced in 
various ways, are used in the workshops. The most direct way of copying a drawing is to trace it on a sheet of tracing paper or tracing cloth, and, 
if more than one copy is required, the ti-acing is used to produce blue prints by sun-printing. 

There are several photo-copying processes used for reproducing copies, or blue prints, by heliography, or svm-printing as it is called, in 
which the tracing is placed in front of, and in close contact with, a sensitized sheet of paper, both being- clamped in a glass frame and exposed to the 
actinic - rays of light which, faUing upon the ti'acing, pass through the transparent portions, decomposing the sensitized paper below, leaving the opaque 
lines upon the tracing undecomposed and transferred to the sensitized sheet. This sheet is then removed from the frame, and washed in water or certain 
solutions to remove the sensitizing matter and thereby develop the lines. 

The sun-printing process has the drawback of being somewhat slow, since it is mainly dependent upon the character of the natural light, and as this 
varies a great deal, so does the time taken to make the prints ; but since the invention some years ago of the electrical photo-copying apparatus, in 
which electricity is used to produce the requisite Ught, engineers have had at theii- command a simple, handy apparatus which makes them independent 
of the weather, and in which prints may be made in two or three minutes. Perhaps the best-known apparatus of this kind is the one invented by 
Messrs. Shaw and Halden, and manufactui-ed by Messrs. J. Halden of Manchester. 

In using the apparatus the tracing and sensitized paper are laid upon a vertical semi-cylindrical glass plate, and a cover or jacket is then laid over 
the back of the sensitized sheet and firmly clamped by engaging with a rod. The cylinder is then turned into position, and the arc lamp lowered and 
raised gradually up and down its interior, the speed being regulated to suit the exposui-e required of various sensitized papers. The lime-light apparatus 
of a lecture lantern can also be used with excellent results for this purpose with a little scheming, but of course it is not so expeditious. 

273. Tracing. — No small amount of skill is required to expeditiously make a good tracing. The beginner cannot do better than commence by 
di-awing a number of straight lines and arcs of different thicknesses on tracing paper and cloth with his drawing pen and bow pen respectively. The ink 

' These are only used in connection with rounded surfaces on machine drawings. 

- The action, as in photography, of the sun's rays in their chemical, as distinct from their illuminating and heating, effects. 



156 



INDUSTRIAL DRAWING AND GEOMETRY 



may be inti-odiiced between the nibs of the pen by a pointed quill, or by a writing pen, care being taken to wipe the outside of the nibs, to prevent any 
ink from them touching the edge of the square, straight edge, or set-square, for should the ink get in contact with these instruments it runs on to the 
paper and spoils the tracing. Care must be taken to preserve uniformity of thickness in the lines where required, and to make arcs and curves 
flow into straight lines without any apparent break, — ia other words, to satisfy the geometrical condition for tangential contact. If the ink 
does not freely run on the tracing paper or cloth, a little powdered chalk may be rubbed over the sheet, or a drop or two of ox-gall may be added 
to the ink. 

In commencing a tracing, be carefvil to pin the tracing paper over the drawing and on to the drawing board in such a way that the sheets are taut, 
and the principal line of the drawing is square with the working edge of the board. As a general rule it is best to draw all the lines that are in the 
direction of the length of the board first. In working down from top to bottom in doing this, many lines will probably be missed, but they will be picked 
up by working down a second, and even a third time if necessary. The transverse lines can then be drawn in the same way, and then any connecting 
arcs drawn, and the circles, if any, described. 

Usually the first tracing made by a beginner is not much of a success, but by persevering in the way indicated he should soon become proficient. 
Great care, of coui-se, has to be taken in writing the dimensions to ensure absolute accuracy. 

274. Tracing Exercises. — You will find that many of the larger diagrams in the figures given in this book are suitable to commence on. After 
a little practice on these, you will be able to attempt the tracing of most of the dra-wing exercises given in the book, of course working on the simpler 
ones first. 

275. Sectional Shading or Lining for Various Materials. — Fig. 524 shows the sectional shading that is very generally used to indicate the 
materials used in engineering work. They speak for themselves. 




IRON 



WROUGHT 
IRON . 



WOOD 



^^ 



Z 



^ 



////■ 



/v/// 



BRICK- 
WORK. 










CONCRETE. STONE. 



HARD FIBRE 
RUBBER. 



Pig. 524. — Conventional sectional lining for various materials. 



CHAPTER XXVI I 

MISCELLANEOUS DRAWING EXERCISES 

1. A hut is shown in Pigs. 525, 526, dimensioned, but not to scale. Draw these two views, and add a plan. Scale i" = 1 foot. 

2. The Figs. 527 to 529 show a cast-iron weight, the plan being incomplete. Draw the complete three views. Scale half full size. 

3. A pictorial view of a hox is shown in Fig. 530. Draw a plan, elevation, and end elevation of it. Full size., 

4. An isometric sketch of a gun-metal block (forming part of a thrust bearing) is shown in Fig. 535. Draw three views of it, making the radius of the arcs li" 
Scale full size. 

5. Draw a pictorial view of the dovetailed joint shown in Figs. 536, 537. 




■///////////////////////////M'/W/yA 



r-0- 



10-6"- 




J 

"5 



S'-6' 







^3' 


a"-* 



-^z'-d'* 



Figs. 527, 528.— Cast-iron weight. Elevation and 
section. 



'////^////M/''W////////////////////////////////////y 



\ 




n r- 


7 


/ 








\ 








"^ „9 - 



Figs. 525, 526. — Elevations of a hut. 



Fig. 529. — Incomplete plan. 



6. The end views (Pigs. 5iO, 541) of two valve cams for petrol motor cam shaft, with their common front elevation (Fig. 542), are shown. Carefully set them 
out. Full size. 

7. A valve rod gland (gun-metal) is shown in plan and elevation in Figs. 548, 544. Draw the two views. Full size. 

8. Make full-size drawings of the cam shaft bearing shown in Figs. 550, 551. Pull size. 

157 



158 



INDUSTRIAL DRAWING AND GEOMETRY 





'/:^ %y,'/i< 



S LA lA U 

2^' ; 



Fig. 530.— Wooden box. 



Fig. 535.— Part of a thrust bearing. 



Figs. 536, 537.— Dovetailed joint. 




-2i- 



FiGS. 588, 589. — Neck bush for stuffing bos. 



^ RADIUS; 




EXHAUST. 



I 16 I .. ^g, 2 I 

•^RADIUS I 

16 , 



NLET. 



Figs. 540-542.— Valve cams. 



MISCELLANEOUS DRAWING EXERCISES 



159 



9. A cast-iron tank flange joint is shown in section (Pig. 547). Assume that the pitch of the bolts is 3", and set out in two views a length of joint that will 
show three bolts. Full size. 

10. Draw, full size, the valve cover for a petrol engine cylinder (Pigs. 548, 549). The hole is threaded to suit the sparking plug. 

11. A petrol engine cylinder cover is shown in Figs. 550, 551. Carefully make the two drawings. Scale full size. 

12. Make a pictorial view of the hut shown in Exercise No. 1. 

13. Show the cast-iron weight (Pigs. 527 to 529) in pictorial projection. Half-size. 

14. Draw two views of the gun-metal stuffing box bush (Pigs. 538, 539), fuU size, and add a sectional plan. 




^ ' 



I" 



■3i-L- 



~T — 



2* 



Figs. 543, 544. — Gun-metal gland. 



CM BUSH 

■3T 



1 STUDS 

16 




Pigs. 545, 546. — Bracket bearing 
for cam shaft. 




Figs. 548, 549.— Valve cover 
for petrol engine cylinder. 




i^-^^^'^^^ 



Mmz^ 



T 
Fig. 547. — Cast-iron flange joint. 



TH"?eoded to 
suit Plua 




Figs. 550, 551. — Petrol engine 
cylinder cover plate. 



DEFINITIONS, ETC. 



You are not exiieoted to laboriously study ,the foUowing definitions, etc. ; they have been arranged, and find a place here, mainly for reference pur- 
poses. But notwithstanding this intention, you will probably read them, and in so doing no doubt learn some things that you will remember, and find 
useful in executing work in this subject. Some of the definitions will be sure to strike you as being self-evident, ones that you instinctively under- 
stand, and would take for granted. However, it wiU do you no harm to see them in print. 

A Point ' (punctuin, " a small hole ") is that which has position, but not magnitude. It is generally indicated by a dot, thus {'), sometimes 
enclosed in a small circle, but can be more accurately represented by two short cross-lines. 

A Line (linea, " a linen thread ") is that which has only length. It has really no thickness or breadth in pure geometry. 

A Straight Line is the shortest distance between its extremities, or is such that, if any two points be taken in it, the part which lies between them 
is the shortest line which can be drawn between those points. For drawing and practical purposes straig'ht lines are drawn or produced with a straight 
edge to guide the pencil, marking or cutting tool. A thin cord stretched by the two ends takes the form of a straight line in plan, and when such a line 
has been rubbed with chalk, and pulled tight over a surface on which a straight line is to be drawn, it can be lifted up at the middle, so as to make it, 
when let go, strike the surface with a little force, and form what is technically called a chalk line. 

Parallel Lines are lines the same distance apart throughout (as in Pig. 552), and which, if produced ever so far both ways, never meet. 



Fig. 552.— Parallel lines. 



An Angle {angulus, " a corner ") is the inclination to each other of two straight lines which meet in a point. The point is called the vertex, and the 
lines meeting in it, sides or legs of the angle. The size of the angle does not depend on the length of the lines, but on their inclination to one another. 




■%<. 





Pig. 554.— Eight angle. 



Pig. 555. — Obtuse angle. 



Pig. 553. — Complement and supplement 
of an angle. 

The Complement of an Angle (Fig. 553) is the angle it requires to complete a right angle. 
The Supplement of an Angle (Fig. 553) is the angle it requires to complete two right angles. 



Pig. 556. — Acute angle. 



ouppiemeni oi an jingie i,r ig. ooo) is uie angle it; requires lo compieie iwo rignt angles. 

Por practical optical purposes, the nearest approach to a geometrical point is the point made by the intersection of two threads of a spider's web. 

160 



DEFINITIONS, ETC. 



161 



Right Angle. — When a straight line standing on another straight line makes the adjacent angles (those on each side of it) eq^ual to one another, 
each of them is called a right angle : and the lines are mutually perpeudicular, and are inclined to one another at an angle of 90° (Pig. 554). 

An Obtuse Angle (obtusus. "blunt") is an angle greater than a right angle (Fig. 555). 

An Acute Angle {actus, ''sharp ") is an angle less than a right angle (Fig-. 556). 

A Circle ' (circuhis, " a ring," " a hoop '") (Fig. 557), is a plane figure contained by one curved line called the circumference, every point of which is 
equally distant from a point within it called the centre.- The curved line itself forming the eireunxference or periphery is also called a circle (or ring). 

NRO 







Pig. 557. — Circumference 
of a circle. 



Pig. 558. — Diameter 
of a circle. 



Fig. 559.— Radius of 
a circle. 



Fig. 560. — Arc of a circle. 



Fig. 561.— Chord of a 
circle. 



The circumference {eircimif evens, ^ " carrying round ") = ir x the diameter. The area of a circle = its diameter^ x ^ nearly ; or = the radius^ X ir 
= 314159, or --f nearly ; therefore j = 0-7864, or \l nearly. 

A Diameter of a Circle {diametros, " a measure through ") is a straight line wliich passes through the centre, arid is terminated both ways 
by the circumference (Fig. 558). 

A Radius of a Circle (radius, "spoke of awheel") is a straight line drawn from the centre to the circumference, and is half a diameter 
(Fig. 5.59). 




Pig. 562. — Segment of a circle. 





Pig. 563. — Semicircle. 



S.ECTOR 



Pig. 564. — Sector of a circle. 




Fig. 565. — Tangent and normal to a circle. 



An Arc of a Circle {arcus, " a bow ") (Fig. 560) is any part of its circvunf erence. 

A Chord (chorde, " a harp-string "') (Fig. 561) is the straight Hue which joins the extremities of an arc. 

A Segment of a Circle (segmentum, " a cutting," " a sHce ") (Fig. 562) is a figure contained by an arc and its chord. 

A Semicircle is haK a circle (Fig. 563). 

• Pythagoras (550 B.C.) discovered that of all figures having the same boundary, the circle among plane figures and the sphere among solids are the most capacious. 
' Kentron, " a goad," " a point." 

' Archimedes (287 B.C.) discovered the relation between circumference and diameter. Befer to the author's " Origin, Rise, and Progress of the Science of 
Geometry," etc., p. 23. 

M 



162 



INDUSTRIAL DRAWING AND GEOMETRY 



A Sector of a Circle (sector, " a cutter ") (Fig. 564) is tte figure contained by an arc and the two radii drawn to its extremities. If the radii be 
at right angles to each other, it is called a quadrant. 

Concyclic. — Points which lie on the same circle are said to be concyclic. 

Area of sector = length of arc x i radius. 

A Tangent (Lat. tango, " to touch ") is a straight line which touches a circle in a point, but which, when produced, does not cut it (Fig. 56.5). 

A Triangle (Lat. tri, " three ; " and angulus, '• a corner "') is a figure which has three sides. (It has been called a trigon.) 

Area of a ti-iang-le = base X J altitude. Centre of gravity is on a centre line from its apex, and \ of its altitude from the base. 








Pig. 566. — Equilateral 
triangle. 



Fig. 567. — Isosceles 
triangle. 



Fig. 568. — Scalene triangle. 



Fig. 569. 



Fig. 570.— Showing 
altitude. 



Fig. 571.— Showing 
altitude. 



An Equilateral Triangle (Lat. sequus, " equal ; " and latus, lateris, " a side ") has three equal sides (Fig. 566). 

An Isosceles Triangle (Gr. isos, " equal ; " and shelos, " a leg ") has two equal sides. The side which is not one of the equal sides is called the base 
(Fig. 567). 

A Scalene Triangle (Gr. scalenos, " uneven") has none of its sides equal (Pig. 568). 

A Right- Angled Triangle contains a right angle. The side opposite the right angle is called the hypotenuse (Gr. hypo, " under " or " beneath ; " 
and teino, " to stretch "), one of the other sides is called the base, and the remaining side the perpendicular or side, these being interchangeable 
according to the position of the triangle (Fig. 569). 

The Vertical Angle of a triangle is the one wliich is opposite the base. 



SQUARE 



RECTANGLE 



RHOMBUS 



RHOMBOID 



Fig. 572.- 



Fig. 573. 



Fig. 574. 



Fig. 575. 





Fig. 576. 



Fig. 577. 



The Altitude or height of a triangle is the perpendicular drawn from the vertical angle to the base or the base produced (Figs. 570, 571). 
Orthocentre. — The intersection of the perpendiculars from corners of a triangle to the opposite .sides. 

A Quadrilateral or Quadrangular Figure (Lat. quatuor, "four ; " latus, lateris, " a side ") is contained by any four straight lines which form a 
closed figure. It is sometimes called a tetragon. 

A Parallelogram (Gr. parallel ; and gramma, " a figure ") is a figure which has the opposite sides equal and parallel to each other. 

A Square is a quadrilateral figure having all its sides equal to each other, and its angles right angles (Fig. 572). 

A Rectangle (Lat. rectus, " right," and angle) (Pig. 573) or oblong has only its opposite sides equal to each other, and all its angles right angles.' 

' The simplest way to test whether the figure is a true rectangle or not, when the opposite sides are equal, is to measure the diagonals, which should be equal. 



DEFINITIONS, ETC. 



163 



A Rhombus (Gr. rhomhos) is a quadrilateral figiu-e which has all its sides equal to each other, hut has no right angles (Fig. .574). 

A Rhomboid (Gr. rhomhos: and eidos. " like ") is a quadrilateral figure which has only its opposite sides equal to each other, and has no right 
angles (Fig. .575). 

A Trapezium (Lai. trapeza, "a table," from ietra, "four;" and jjeza, "foot'') is a quadrilateral figure, the opposite sides of which are neither 
parallel nor equal (Fig. .576). 

A Trapezoid (Gr. trapezian ; and eidos, " like ") is a quadrilateral figure, two of its sides being parallel, but none equal (Fig. .577). 

A Diagonal (Gr. dia, "through; " and gonia, "an angle ") of a straight-lined figure is a straight line joining opposite angiilar points. 









Pig. 578. 



Fig. 579. 



Fig. 580. 



Fig. 581. 



Fig. 582. 



Fig. 583. 



A Polygon (Gr. poly, " many ; " and gonia, " angle ") is a figui-e which has more than four sides. When aU its sides are equal, a polygon is said to 
be equilateral. A polygon is equiangular when all its angles are equal. It is sometimes refeiTed to as a multilaieral figure. 

A Regular Polygon is both equilateral and equiangular. 

An Irregular Polygon has its sides and angles unequal. It may have such a form that all its external angles are prominent or salient (Lat. salio, 
" to leap "), or one or more external angle may re-enter or be re-entrant. Thus, in the irregular polygon (Fig. 129), the exterior angles at A, B, D, E, 
F, are salient, but the angle at C is re-entrant. 

SiJecial names are given to polygons to indicate the number of sides. For example : — 

A Pentagon (Gr. pente, " five ; "' and gonia, " an angle '') is a figure of five sides (Fig. 578). 

A Hexagon (Gr. hex, " six ; '' and gonia, " an angle ") is a figui-e of six sides (Fig. 579). 









Fig. 584. 



Fig. 585. 



Fig. 586.— Triangles of 
equal area. 



Fig. 587.— Annulus. 



Fig. 588. —Eccentric Fig. 588a. — Circumscribed 
annulus. and inscribed circles. 



A Heptagon (Gr. hepta, " seven ; " and gonia. " an angle ") is a figure of seven sides (Fig. 580). 

An Octagon (Gr. octo, " eight ; " and gonia, " an angle ") is a figiu-e of eight sides (Fig. 581). 

A Nonagon (Lat. nomis, " ninth ; '' and gonia, " an angle ") is a figure of nine sides (Fig. 582). 

A Decagon (Gr. deha, " ten ; " and gonia, " an angle ") is a figure of ten sides (Fig. 58.3). 

An Undecagon (Lat. undecini, " eleven ; " and Gr. gonia, "an angle ") is a figure of eleven sides (Fig. 584). 

A Duodecagon (Lat. duo, " two ; " Gr. deka, " ten ; " and gonia, " an angle ") is a figiu-e of twelve sides (Fig. 585). 

Circumscribed and Inscribed Circles. — Fig. 588a shows an equilateral triangle circumscribed and inscribed. 



164 INDUSTRIAL DRAWING AND GEOMETRY 

The Perimeter (Grr. peri, " around ; " and metron, " that which measures ") of a plane iigtire is the sum of all its sides, or its boundary. 

A Proposition (Grr. ^ro, " before ; " and pono, " to place ") is that which is offered or proposed for adoption or consideration. 

Propositions (for geometrical purposes) are of two kinds, viz. problems and theorems. 

A Problem (Gr. pro, "before ; " and ballo, " to throw ") is a proposal to do a thing, such as to solve a question, or to draw a figure. 

A Theorem (Gr. iheorema, " something which can be seen," literally, " a sight ; " median, " traversing the middle, lengthwise ") is a proposition to be 
proved by a certain chain of reasoning. In a theorem some new principle is asserted to be true, the truth of which is almost seK-evident. 

A Corollary is usually defined as a statement the truth of which follows readily from an established proposition ; it is therefore appended to the 
proposition as an inference or deduction, which u.sually requires no further proof. 

Congruent Figures. — Figures which are equal in all respects are said to be congruent (Lat. congruo, "to agree"). 

Superposition. — If two figiu-es when applied to or laid over one another can be made to fit exactly, or coincide, they must be equal in all 
respects, and this method of testing equality is known as the method of superposition. 

Equivalent Figures. — Figures which are equal in area (but not necessarily congruent) are said to be equivalent. 

The Apothem of a regular polygon is the perjjendicular from the centre to the side. 

Definitions, and Summary of some Useful Particulars relating to Areas. 

A knowledge of the arithmetical measure of an area may often enable you to solve with facility a problem on areas ; therefore the measures of 
areas of varioi.iS fignres which follow should be found useful for reference. 

Area. Definition. — The boundary-line or perimeter of any closed figure encloses an amount of surface called its area. 
Area of Rectangle = length x breadth. 

,, parallelogram = length of side x distance between sides. 

,, triangle = base B x i perpendicular height H. Fig. 586. (Euc. I. 377.) 

„ any regular polygon = radius of inscribed circle x number of sides x i length of one side. 

■ 1 T 9 T i ■> "^ T> .-, nnr- ■ T T ., i. area of a circle 31 22 11 

„ circle = radius'' X IT, or diameter- x ,= d^ 0-7854 nearly, and the ratio j— . .^^ -- = -r = s6 = T7- 

4 *" area ot circumscribing square 4 28 14 

„ sector of circle = radius x h length of arc or 

no. of dea-rees in arc „ ., . , 

— §^ X area ot the circle. 

„ segment of circle = area of the sector — j chord x (radius — versin).' 

the ring, or annulus = 7r(R2 - r^). (Figs. 5'87, 588.) 

„ ellipse = major axis x minor axis x 0'7854. 

„ parabola = base x f height. 

„ surface of sphere = diameter x tt = (§ surface of circumscribing cylinder). 

„ cylinder = (length x circumference) + area of both ends. 

„ cone = (circumference of base x \ slant height) + area of base. 

„ frustum of cone = (sum of circumferences at both ends x ^ slant height) + area of both ends. 

Definitions, etc., relating to Solids and their Projection. 

These are given not necessarily for systematic study, but rather for reference purposes. Many of them are in common use in the 
science of projection. 

Solids are aU bodies that have the three dimensions, leng'th, breadth, and thickness; they have an infinite variety of shape, some being- bounded 

• The versin is the perpendicular distance between the chord and arc. 



DEFINITIONS, ETC. 



165 



by curve surfaces, and some by plane surfaces, whilst o-thers are bounded by a combination of sucb surfaces. The foUowing- particulars of some of 
the principal solids will give examples of each kind. In geometrical language those that are terminated or bounded by regular and equal similar 
planes are called regular solids, such as the tetrahedron, cube, octahedron, dodecahedron, and icosahedron. (These are called the five regular solids.) 
All the regular solids can be inscribed in or made to circumscribe a sphere. 

The Tetrahedron (Gr. tetra, " four ; " and hedra, " a side ") (Fig. 589), one of the five regular solids, is a triangular pyramid bounded by four equal 
equilateral triang-les. 

The Cube (Gr. Icyhos. "a die") (Fig. 590) is one of the five regular solids, consisting of or bounded by six equal square bases or sides, and 
its angles are all right angles. 

The Octahedron (Gr. olcto, "eight;" and hedra, "a side") (Fig. 591) is one of the five regular solids, bounded by eight equal equilateral 
triangles. 









Fig. 589.— Tetrahedron. 



Fig. 590.— Cube. 



Fig. 591.— Octahedron. Fig. 592.— Dodecahedrou. Fig. 598.— Icosahedi-on. Fig. 594.— Sphere, 

and hedra, " a base or side ") (Fig. 592) is one of the five regular soKds ; it is bounded by twelve 



The Dodecahedron (Gr. dodeha, " twelve 
regular pentagons as faces. 

The Icosahedron (Gr. eihosi, "twenty;" and hedra, '-a base or side") (Fig. 593) is one of the five regular solids; it is bounded by twenty 
equilateral triangles. 

A Polyhedron is any solid bounded by plane figures. 



SURFACES AND VOLUMES OF THE FIVE REC^ULAE SOLIDS (Edges = 1). 



Name of Solid. 


Sarface. Volume. 


Tetrahedron .... 
Cube or Hexahedron . 
Octahedron .... 
Dodecahedron . . . 
Icosahedron .... 


1-7320508 01178511 
6-0000000 1-0000000 
3-4641016 0-4714045 
20-6457288 7-6631189 ' 
8-6602540 2-1816949 



The Sphere' (Gr. .s^j/io/m, " a ball") (Fig. 594) is a solid contained within one uniform surface, every point of which is equally distant from a point 
within called the centre, and may be conceived to be generated by the revolution of a semicircle about its diameter, which is fixed. All sections 

' Archimedes (287 B.C.) discovered that the solidity and surface of the sphere are | of the circumscribing cylinder. Refer to the author's " Origin, Rise, and 
Progress of the Science of Geometry," etc., p. 23. 



J 66 



INDUSTRIAL DRAWING AND GEOMETRY 



of a sphere are circles. If a spliere be cut by a plane passing tliroug-h it, each part is called a segment. When the cutting plane passes through the 

centi-e. each part is a hemispliere ; any part cut oif between two planes is called a zone. 

(The area of a sphere's surface = ?r x its diameter'. The capacity of a spliere = 0-.5236 x its diameter-.) 

A Circular Spindle (Pig. 59-5) is a solid that may be conceived to be formed by the revolution of a circular arc or segment ABC about its chord 

AB, which remains fixed. 

A spheroid (or ellipsoid) (Fig. 596) is a solid that may be conceived to be formed or generated by the revolution of an ellipse about one of its axes. 

If the revolution be made about the major axis, the solid is called a prolate spheroid ; but if about the minor axis, an oblate spheroid 

A Right ' Cylinder (Grr. kylindros, " a roUer ") (Fig-. 597) is a solid bounded by one circular and two plane surfaces, and may be conceived to be 

formed or o-enerated by the revolution of a rectangle about one of its sides, which is fixed, and is called the axis of the cylinder. Either of the two plane 

surfaces is "called the base of the cylinder. When the axis is inclined to the bases, it is an oblique cylinder. 

(Area of a cylinder's surface = area of both ends + length x circumference. Capacity = area of one end x length.) 

A Prism- {Grv. pn'sma. from 2^nzo, "to saw") is a solid whose two ends are any plane figm-es which are equal, similar,^ and parallel, and its sides 

parallelograms. Its axis is the straight line joining the centi-es of its ends or bases. 







Fig. 595.— Circular spindle. Fig 596.— Spheroid. 



Fig. 597.— Cylinder. ■ Fig. 598.— Triangular 

prism. 



iffiP' 

Fig. 599.— Square 
pyramid. 




A Right ' Prism (Fig. 598) is one liaving its axis perpendicular to its ends. The one shown is a triangular prism. When the axis is inclined to its 
ends, it is an oblique prism. 

A Cuboid is a cube-like solid, such as a square or rectangular slab (Fig. 601). 

A Pyramid (Egyptian word) (Fig. 599) is a solid whose base is a polygon, and whose sides are triangles, their apices meeting in one point called 
the apex or vertex of the pyramid. When the axis is inclined to the base it is an oblique pyramid. 

(Capacity of a pyramid = area of base x i perpendicular height. Its e.g. (centre of gravity) is on the axis, and ^ height from base.) 

A Right ^ Pyramid is one having its axis perpendicular to its base. The one shown is a square pyramid (Fig. 599). 

A Cone (Fig. 600) (Gr. Iconos) is a solid having a circular base, and its other extremity terminating in a single point called its apex or vertex. 
It may be conceived to Ije generated by the revolution of a right-angled triangle about one of its sides containing the right angle, which is fixed, and 
is called the axis of the cone. It is sometimes convenient to consider it as a pyi-amid with an infinite niunber of sides, and its base as a polygon 
with an infinite number of sides (that is to say, a circle). When its axis is inclined to the base, it is an oblique cone. 

(Area of surface of a Cone = area of base -f circumference of base x i slant height. Capacity = area of x J perpendicular height. Its e.g. 
(centre of gravity) is on the axis, J height from base.) 

' It is said to be a right cylinder when the axis is perpendicular to its bases or ends. 

- If a solid be terminated by two dissimilar parallel planes as ends, and the remaining surfaces joining the ends be also planes, the solid is called a prismoid. 
^ As you have seen, figures that are similar and equal are called congruent or identical. For much interesting information relating to the properties of 
congruent figures, refer to Prof. Henrici's " Geometry of Congruent Figures." 
■" It is said to be right when its axis is perpendicular to its bases or ends. 




DEFINITIONS, ETC. 167 

A Parallelepiped (Fig. 601) (Gr. jjocoZZe/. epi. " upon ; " and pedmi, " the ground ") is a regular solid Ijounded by six parallelograms, tlie ojiposite 
ones of wliich are equal and parallel, or it is a prism whose base is a parallelogram. Thus a brick is a parallelepiped. These solids are sometimes called 
cuboids. 

Frustum (Lat. " a piece," " a bit "). — The frustum of a coue, pyramid, or other solid is the part near the base formed by cutting 
off the top. It is said to be truncated. 

Volume of a Solid. — As in the case of areas of plane figures, the number of solids whose volume can be calculated from their 
linear dimensions is very limited. But by immersing a solid completely in a liquid, it displaces a quantity of that liquid whose 
volume is the same as that of the solid. This gives us a simple means of flndiag such volumes, and obviously we eau in tliis way ''■ °^ ■ (juboid. 
also find the internal volume of a vessel of any shape, as the engineer sometimes does in measuring the volum.e of the steam-passages and clearance 
space of a steam-cylinder. 

Projections. — The plan and elevation of an object are called its projections. The projection of a point on a plane is the extremity of the prelector 
let fall from the poittt to the plane. 

When a line, or the plane surface of a solid, is parallel to a plane, its projection on that plane is equal to the line or siu-face, and the greater the 
inclination the shorter becomes its projection, the limit occurring' when the line or sin-face is perpendicular to the plane, when its projection becomes a 
point or line respectively. The conventional way to represent or name the corners of an object is by capital letters, and then- plans by small itaKcs, 
and their elevations by the same italics with dashes. 

Projectors (Ij&i. pro, •■forward;" &uAjacio. jadum, "to throw"). — The lines connecting the in'ojections of an object, or lines from points on an 
object to the corresponding points on the projection, are called projectors. 

Section (Lat. seco, " to cut "). — When a body is cut by a plane,' the STU-face or shape of the cut part or sui-face of separation is caUed a section. 

Sectional Elevation. — If the section of a body be di-awn in elevation, and the parts attached to it, but not cut by the section plane, be shown, the 
view is called a sectional elevation. 

Orthographic (Gr. orthos, " straight ; " and graplio, " to wi-ite "). — Pertaining to orthography, which in geometry refers to the projections of objects 
showing all the parts thereof in their true proportions. 

Orthogonal (Gr. orthos, and gouia, " an ang'le "). — Right-angled, from ortliogon, " a rectangular figure." Thus we speak of the planes of projection 
as being orthogonal planes. In the ease of an ordinary projection, the projections are perijendicular to the planes of projection, and the system is 
said to be orthographic ; but when the projectors are all equally inclined at any other angle to the planes, as in the case of shado^ws, the system is 
referred to as orthogonal. 

A Plane (Lat. planus, "level," "flat") is a plain level surface. If any two points be taken in a true plane, the line joining the points will be wholly 
in the plane. The term " plane " is often used to express an imaginary surface. 

Co-ordinate Planes. — The horizontal and vertical planes (H.P. and V.P. respectively) of projection ajre called the co-ordinate planes of projection, 
and their intersection is represented by the letters XT. 

The Traces of a Plane are the lines made by the plane cutting the co-ordinate planes. 

Inclined Planes are planes that are inclined to the horizontal, plane, and are also perpendicular to the vertical plane. 

Oblique Planes (Lat. obliquus, "slanting") are planes that are inclined to both the horizontal and vertical planes. They are, therefore, inclined 
planes which are not perpendicular to the vertical plane. Strictly speaking, all inclined planes are oblique planes. 

Tangent Planes. — When a plane touches a sphere in a point on its surface, it is said to be tangent to it. Similarly, if a plane touches the sui-face 
of a cone in a Hne passing tlu-ough its apex, the plane is called a tangent plane ; and, again, if a plane touches the surface of a cylinder in a line parallel 
to its axis, it is tangent to the solid. 

Constructed. — A plane is said to be constructed when it has been folded or revolved about its trace (as an axis) into a plane of projection, cai-rying- 
with it all figui'es, Hnes, and points wliioh are contained in it. 

Locus (Lat. "a place"). — When a jilane is being constructed, say, about its horizontal ti-ace, the successive plans of any point in the plane form 
a line perpendicular to the horizontal trace. Tliis line is called the locus of the plan of the point. Loci is the plural of locus. 

' This is a geometrical expression. Of course the body is not literally cut. A line or plane is used to represent the position of a out. 



168 INDUSTRIAL DRAWING AND GEOMETRY 

Horizontal (G-r. horos, " a boundary ")• — A line or plane is said to be horizontal when it is parallel to the horizon, or to the surface of still water, 
or perpendicular to the direction of a plumb-line at any point. 

Horizontal Trace. — The point in which any line or line produced pierces the horizontal plane is called its liorizontal trace ; similarly, the line in 
which any plane cuts the horizontal plane is called the liorizontal trace of the plane. 

Vertical (Lat. verto, "to turn"). — A line or plane placed perpendicular to the horizontal plane is vertical (the plumb-line used by bricklayers and 
others always hang-s in a vertical line). 

Vertical Trace. — The point in which any line or line produced pierces the vertical plane is called its vertical trace ; similarly, the line in which 
any plane cuts the vertical plane is called the vertical trace of the plane. 

Hidden Lines. — The projections of hidden lines are drawn dotted. 

Contiguous. — Two faces of a solid are said to be contiguous when they intersect or are adjacent. 

Dihedral Angle (Gr. di, " two ; " and hedra, " a side "). — The true ang-le formed by the intersection of two planes is termed a dihedral angle. 

Trihedral Angle (Gr. tri, " tlu-ee ; "' hedra, " a side "). — The solid ang-le of a solid formed by the intersection of three of its sides in a corner. 



TABLES OF BRITISH AND METRICAL EQUIVALENTS 



1.— LENGTH 



English to Meteioal 

1 inch = 25'4 millimetres = 2-54 centi- 
metres 
1 foot = 30-4799 centimetres 
1 yard = 0-914399 metre 
1 chain = 66 ft. = 20-1168 metres 
1 mile = 5280 ft. = 80 chains 
0-62137 mile = 1 kilometre 



Metrical to English 

inches : 



1 millimetre = 0-03937 
2 -.5" 
nearly, or -^r- nearly 

1 centimetre = 10 mm. = 0-8937 inch 
= a full I" 

( 39-37 inches = 39§" nearly 
1 metre = 3-280843 feet 

I 1-093614 yards 
1 kilometre = 1000 metres = 3280-9 
feet 



3.— LAND MEASURE. 



7-92 inches 

100 links or 66 feet 

10 chains 

80 chains, or 8 furlongs 
9 square feet 
30i square yards 

16 perches 

40 perches 
4 roods 
640 acres 

30 acres 
100 acres 



= lliiik 

= 1 chain (Gunter's) 

= 1 furlong 

= 1 mUe 

= 1 square yard 

= 1 square rod, pole, or perch 

= 1 square chain 

= 1 rood 

= 1 acre 

= 1 square mile 

= 1 yard of land 

= 1 hide of land 



The side of a square whpse area is one acre is equal to 208-71 feet. 



2.— SURFACE AND AREA 



English to Metrical 

1 sq. inch = 6-4516 sq. centimetres 
1 sq. foot = 929-03 sq. centimetres 

= 0-092903 sq. metre 
1 sq. yard = 0-836126 sq. metre 
1 acre = 0-40468 hectare 
1 sq. mile = 259 hectares 



Metrical to English 

1 sq. centimetre = 0-155 sq. inch 
I sq. metre = 10-7639 sq. feet 
,, = 1-196 sq. yards 

100 sq. metres = 1 are 
1 hectare = 100 ares = 10,000 
metres = 2-4711 acres 



sq. 



The Square is used in measuring roofing and flooring. It equals 100 
sq. feet. The Eod is used in measuring brickwork. It equals 272 super, feet 
l| brick thick = llj ou. yards = 306 cu. feet. 



4.— VOLUME 



English to Meteioal 

1 cu. inch = 16-387 cu. centimetres 
1 cu. foot = 0-028317 cu. metre 

= 28-317 litres 
1 cu. yard = 0-764553 cu. metres 

= 764-553 litres 
1 gaUon = 4-545963 Utres 
= 0-1605 cu. feet 
,, = 277-27 cu. inches 



Metrical to English 

1 cu. centimetre = 0-061 cu. inch 
1 cu. decimetre = 61-024 cu. inches 
1 litre = 1000 cu. centimetres 

1-7598 pints 
1 cu. metre = 35-3148 ou. feet 

„ = 1-307954 ou. yards 



1 U.S.A. gallon = 0-83254 Imperial gallon = 231 cu. inches. 



MISCELLANEOUS CONSTANTS 



One Radian = 57-30 degrees 

1 Gallon = 0-1604 cubic foot = 10 lbs. of water at 62° P. 

l.Knot = 6080 feet per houj = 1 Nautical mile per hour 



Weight of 1 lb. in London = 446,000 dynes 

1 lb. avoirdupois = 7000 grains = 453-6 grammes 

1 Cubic Foot of Water weighs 62-3 lbs. 



THE END. 



169 



PRINTED BV 

WILLIAM CLOWES AND SONS, LIMITED, 

LOKDON AND BECCLES 



